Solvable group: Difference between revisions

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{{pivotal group property}}
[[importance rank::2| ]]
 
{{semibasicdef}}
 
{{finite-at|finite solvable group}}
 
==History==
 
{{term introduced by|Galois}}
 
The notion of ''solvable group'' arose from the attempt to characterize the Galois groups of those field extensions which could be solved by means of radicals.
 
In fact, the term ''solvable'' arose precisely because a normal field extension is contained in a radical extension if and only if the Galois group is solvable.
 
==Definition==
==Definition==


===Symbol-free definition===
'''Solvable''' is also called '''soluble''' by some people.


A [[group]] is said to be '''solvable''' if any of the following equivalent conditions holds:
===Equivalent definitions in tabular format===


* There is a [[normal series]] of finite length starting from the trivial subgroup and ending at the whole group with each successive quotient being an [[Abelian group]]
{| class="sortable" border="1"
* There is a [[subnormal series]] of finite length with each successive quotient being an [[Abelian group]]
! No. !! Shorthand !! A group is termed solvable if ... !! A group <math>G</math> is termed solvable if ...
* The [[derived series]] reaches the identity in finitely many steps. The number of steps needed is termed the [[solvable length]] of the solvable group.
|-
| 1 || normal series, abelian quotients || there is a [[defining ingredient::normal series]] of finite length starting from the trivial subgroup and ending at the whole group with each successive [[quotient group]] being an [[defining ingredient::abelian group]]. || there exists a series of subgroups <math>\{ e  \} = H_0 \le H_1 \le \ldots \le H_n = G</math> such that each <math>H_i</math> is [[normal subgroup|normal]] in <math>G</math> and each <math>H_{i+1}/H_i</math> is [[abelian group|abelian]].
|-
| 2 || subnormal series, abelian quotients || there is a [[defining ingredient::subnormal series]] of finite length starting from the trivial subgroup and ending at the whole group with each successive [[quotient group]] being an [[abelian group]]. || there exists a series of subgroups:<math>\{ e \}= H_0 \underline{\triangleleft} H_1 \underline{\triangleleft} \dots \underline{\triangleleft} H_n = G</math> such that each <math>H_i</math> is normal in <math>H_{i+1}</math> and each <math>H_{i+1}/H_i</math> is [[abelian group|abelian]].
|-
| 3 || derived series finite length || the [[defining ingredient::derived series]] reaches the identity in finitely many steps || the [[derived series]] of <math>G</math>, i.e., the series <math>G^{(n)}</math> where <math>G^{(0)} = G</math> and <math>G^{(i+1)} = [G^{(i)}, G^{(i)}]</math> is the [[defining ingredient::derived subgroup]] of its predecessor, reaches the trivial subgroup in finitely many steps.
|-
| 4 || characteristic series, abelian quotients || there is a [[defining ingredient::characteristic series]] of finite length starting from the trivial subgroup and ending at the whole group with each successive [[quotient group]] being an [[defining ingredient::abelian group]]. || there exists a series of subgroups <math>\{ e  \} = H_0 \le H_1 \le \ldots \le H_n = G</math> such that each <math>H_i</math> is [[characteristic subgroup|characteristic]] in <math>G</math> and each <math>H_{i+1}/H_i</math> is [[abelian group|abelian]].
|-
| 5 || fully invariant series, abelian quotients || there is a [[defining ingredient::fully invariant series]] of finite length starting from the trivial subgroup and ending at the whole group with each successive [[quotient group]] being an [[defining ingredient::abelian group]]. || there exists a series of subgroups <math>\{ e  \} = H_0 \le H_1 \le \ldots \le H_n = G</math> such that each <math>H_i</math> is [[fully invariant subgroup|fully invariant]] in <math>G</math> and each <math>H_{i+1}/H_i</math> is [[abelian group|abelian]].
|}


===Definition with symbols===
The length of the [[derived series]], and the smallest possible length of a series for any of the other equivalent definitions, is termed the [[derived length]] or solvable length of the group.
{{tabular definition format}}


A [[group]] <math>G</math> is said to be '''solvable''' if it satisfies any of the following equivalent conditions;
===Equivalence of definitions===


* There exists a series of subgroups:
{{further|[[Equivalence of definitions of solvable group]], [[equivalence of definitions of derived length]]}}


<math>e = H_0 \le H_1 \le \ldots \le H_n = G</math>
==Examples==


such that each <math>H_i</math> is [[normal subgroup|normal]] in <math>G</math> and each <math>H_i/H_{i-1}</math> is [[Abelian group|Abelian]].
{{group property see examples}}


* There exists a series of subgroups:
===Extreme examples===


<math>e = H_0 \triangleleft H_1 \triangleleft \ldots \triangleleft H_n = G</math>
* The [[trivial group]] is solvable.
* [[Symmetric group:S3]] is the smallest solvable non-abelian group.


such that each <math>H_{i-1}</math> is normal in <math>H_i</math> and each <math>H_i/H_{i-1}</math> is [[Abelian group|Abelian]].
===Groups satisfying the property===


* The [[derived series]] of <math>G</math>, viz the series <math>G^{(n)}</math> where <math>G^{(0)} = G</math> and <math>G^{(i+1)} = [G^{(i)}, G^{(i)}]</math>, reaches the trivial subgroup in finitely many steps.
{{groups satisfying property by importance rank}}


===Equivalence of definitions===
===Groups dissatisfying the property===


{{further|[[Equivalence of definitions of solvable group]]}}
{{groups dissatisfying property sorted by importance rank}}


==Formalisms==
{{semibasicdef}}
{{pivotal group property}}
{{finite-at|finite solvable group}}


{{group extension operator}}
==Metaproperties==
The group property of being solvable can be obtained in either of these equivalent ways:
 
* By applying the [[poly operator]] to the group property of being [[Abelian group|Abelian]]
* By applying the [[finite normal series operator]] to the group property of being [[Abelian group|Abelian]]
* By applying the [[finite characteristic series operator]] to the group property of being [[Abelian group|Abelian]]
 
Note that all these three operators have the same effect in the case of Abelian groups, though in general they may not have.
 
==Examples==
 
For a complete listing of important examples of solvable groups, refer [[:Category:List of solvable groups]].


Particular note-worthy examples are given below:
{| class="sortable" border="1"
 
! Metaproperty name !! Satisfied? !! Proof !! Statement with symbols
* [[Symmetric group:S3|The symmetric group on 3 elements]] is the smallest solvable non-[[Abelian group|Abelian]](in fact, non-[[nilpotent group|nilpotent]]) group
|-
* [[Symmetric group:S4|The symmetric group on 4 elements]] is also solvable
| [[satisfies metaproperty::pseudovarietal group property]] || Yes || [[solvability is pseudovarietal]] || Solvability is closed under taking subgroups, quotients, and finite direct products (more below).
* The [[dihedral group]] of any order is solvable, Further it is nilpotent only when the order is a power of 2.
|-
| [[satisfies metaproperty::extension-closed group property]] || Yes || [[solvability is extension-closed]] || Suppose <math>H</math> is a [[normal subgroup]] of <math>G</math> such that both <math>H</math> and the [[quotient group]] <math>G/H</math> are [[solvable group]]s. Then <math>G</math> is a [[solvable group]].
|-
| [[satisfies metaproperty::subgroup-closed group property]] || Yes || [[solvability is subgroup-closed]] || If <math>G</math> is solvable, and <math>H \le G</math> is a [[subgroup]], then <math>H</math> is solvable.
|-
| [[satisfies metaproperty::quotient-closed group property]] || Yes || [[solvability is quotient-closed]] || If <math>G</math> is solvable, and <math>H</math> is a [[normal subgroup]] of <math>G</math>, the [[quotient group]] <math>G/H</math> is solvable.
|-
| [[satisfies metaproperty::finite direct product-closed group property]] || Yes || [[solvability is finite direct product-closed]] || If <math>G_1, G_2, \dots, G_n</math> are solvable, the [[external direct product]] <math>G_1 \times G_2 \times \dots \times G_n</math> is also solvable.
|-
| [[satisfies metaproperty::finite normal join-closed group property]] || Yes || [[solvability is finite normal join-closed]] || If <math>G</math> is a group and <math>N_1,N_2,\dots,N_r</math> are all [[solvable normal subgroup]]s of <math>G</math>, the [[join of subgroups]] (in this case also the [[product of subgroups]]) <math>N_1N_2\dots N_r</math> is also solvable.
|-
| [[satisfies metaproperty::isoclinism-invariant group property]] || Yes || [[isoclinic groups have same derived length]] || If <math>G_1</math> and <math>G_2</math> are [[isoclinic groups]], then <math>G_1</math> is solvable if and only if <math>G_2</math> is. Moreover, if so, the derived length of <math>G_1</math> equals the derived length of <math>G_2</math>, unless one of the groups is trivial and the other is nontrivial abelian.
|}


==Relation with other properties==
==Relation with other properties==
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===Stronger properties===
===Stronger properties===


* [[Nilpotent group]]: {{proofat|[[Nilpotent implies solvable]]}}
{| class="sortable" border="1"
* [[Supersolvable group]]: {{proofat|[[Supersolvable implies solvable]]}}
! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions !! Comparison
* [[Polycyclic group]]: {{proofat|[[Polycyclic implies solvable]]}}
|-
| [[Weaker than::abelian group]] || [[derived subgroup]] is trivial || [[abelian implies solvable]] || [[solvable not implies abelian]] {{strictness examples|solvable group|abelian group}} || {{intermediate notions short|solvable group|abelian group}} ||
|-
| [[Weaker than::cyclic group]] || || || {{strictness examples|solvable group|cyclic group}} || {{intermediate notions short|solvable group|cyclic group}} ||
|-
| [[Weaker than::nilpotent group]] || [[lower central series]] reaches the identity || [[nilpotent implies solvable]] || [[solvable not implies nilpotent]] {{strictness examples|solvable group|nilpotent group}} || {{intermediate notions short|solvable group|nilpotent group}} || [[nilpotent versus solvable]]
|-
| [[Weaker than::metabelian group]] || [[abelian normal subgroup]] with abelian quotient; derived length two || || {{strictness examples|solvable group|metabelian group}} || {{intermediate notions short|metabelian group|solvable group}} ||
|-
| [[Weaker than::supersolvable group]] || [[normal series]] with cyclic factor groups|| [[supersolvable implies solvable]] || [[solvable not implies supersolvable]] {{strictness examples|solvable group|supersolvable group}} || {{intermediate notions short|solvable group|supersolvable group}} ||
|-
| [[Weaker than::polycyclic group]] || [[subnormal series]] with cyclic factor groups<br>equivalent to solvable in the [[finite solvable group|finite case]]  || [[polycyclic implies solvable]] || [[solvable not implies polycyclic]] {{strictness examples|solvable group|polycyclic group}} || {{intermediate notions short|solvable group|polycyclic group}} ||
|-
| [[Weaker than::metacyclic group]] || [[cyclic normal subgroup]] with cyclic quotient group || || {{strictness examples|solvable group|metacyclic group}} || {{intermediate notions|solvable group|metacyclic group}} ||
|}


===Weaker properties===
===Weaker properties===


* [[HypoAbelian group]]
{| class="sortable" border="1"
* [[Imperfect group]]
! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions !! Comparison
|-
| [[Stronger than::hypoabelian group]] || transfinite derived series reaches identity;<br>equivalent to solvable in the [[finite solvable group|finite case]] || [[solvable implies hypoabelian]] || [[hypoabelian not implies solvable]] || {{intermediate notions short|hypoabelian group|solvable group}} ||
|-
| [[Stronger than::imperfect group]] || no nontrivial perfect quotient group || [[solvable implies imperfect]] || [[imperfect not implies solvable]] || {{intermediate notions short|imperfect group|solvable group}} ||
|-
| [[Stronger than::locally solvable group]] || every finitely generated subgroup is solvable<br>equivalent to solvable in the [[finite solvable group|finite case]] || || || ||
|-
| [[Stronger than::residually solvable group]] || every non-identity element has a non-identity image in some solvable quotient<br>equivalent to solvable in the [[finite solvable group|finite case]] || || || ||
|}


===Conjunction with other properties===
===Conjunction with other properties===


* [[Solvable T-group]] is the conjunction with [[T-group]]
{| class="sortable" border="1"
* [[Solvable HN-group]] is the conjunction with [[HN-group]]
! Conjunction !! Other component of conjunction !! Additional comments
|-
| [[finite solvable group]] || [[finite group]] || For finite groups, being solvable is equivalent to being polycyclic, and has many other alternative characterizations.
|-
| [[solvable T-group]] || [[T-group]] ||
|-
| [[solvable HN-group]] || [[HN-group]] ||
|}


==Metaproperties==
==Formalisms==


{{Extension-closed}}
{{group extension operator}}
The group property of being solvable can be obtained in either of these equivalent ways:


The group property of being solvable is idempotent with respect to the [[group extension operator]]. In other words, if a group has a solvable normal subgroup, and the quotient group is solvable as an abstract group, then the whole group is solvable.
* By applying the [[poly operator]] to the group property of being [[abelian group|abelian]]
* By applying the [[finite normal series operator]] to the group property of being [[abelian group|abelian]]
* By applying the [[finite characteristic series operator]] to the group property of being [[abelian group|abelian]]


In fact, we can just take a subnormal series corresponding to the normal subgroup and pull back a subnormal series corresponding to the quotient group, and put the two subnormal series together to obtain a subnormal series for the whole group.
Note that all these three operators have the same effect in the case of abelian groups, though in general they may not have.
 
{{Quasivarietal}}
 
The property of being solvable of [[solvable length]] at most <math>d</math>, is a [[varietal group property]] -- it is in fact equationally defined by the vanishing of the commutator of any <math>d</math> elements. From this, we can deduce that the group property of being solvable is quasivarietal:
 
* Any subgroup of a solvable group is solvable, in fact, with the same (or smaller) solvable length
* Any quotient of a solvable group is solvable, in fact, with the same (or smaller) solvable length
* Any finite direct product of solvable groups is solvable, in fact, with solvable length bounded by the maximum of the solvable lengths of the groups
 
{{proofat|[[Solvability is quasivarietal]]}}
 
{{S-closed}}
 
Any subgroup of a solvable group is solvable. This follows from its being quasivarietal. See above.
{{Q-closed}}
 
Any quotient of a solvable group is solvable. This follows from its being quasivarietal. See above.
 
{{finite DP-closed}}
 
A finite direct product of solvable groups is solvable. This follows from its being quasivarietal. See above.
 
{{finite N-closed}}
 
A join of finitely many [[solvable normal subgroup]]s is also solvable.


==Testing==
==Testing==
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The problem of testing whether a group is solvable or not reduces to the problem of computing its [[derived series]]. This can be done when the group is described by means of a generating set, if the [[normal closure]] algorithm can be implemented.
The problem of testing whether a group is solvable or not reduces to the problem of computing its [[derived series]]. This can be done when the group is described by means of a generating set, if the [[normal closure]] algorithm can be implemented.


{{GAP command for gp|IsSolvable}}
{{GAP command for gp|
test = IsSolvableGroup}}


To determine whether a group is solvable or not, we cna use the following GAP command:
To determine whether a group is solvable or not, we cna use the following GAP command:


<pre>IsSolvable (group);</pre>
<pre>IsSolvableGroup(group);</pre>
 
where <tt>group</tt> may be a definition of the group or a name for a group previously defined.


where <pre>group</pre> may be a definition of the group or a name for a group previously defined.
==Study of this notion==
==Study of this notion==


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Also closely related is 20F19: Generalizations of nilpotent and solvable groups.
Also closely related is 20F19: Generalizations of nilpotent and solvable groups.


==External links==
==References==
{{searchbox|"solvable+group"}}
===Textbook references===
===Definition links===


* {{wp|Solvable_group}}
{| class="sortable" border="1"
* {{planetmath|Solvable}}
! Book !! Page number !! Chapter and section !! Contextual information !! View
* {{mathworld|SolvableGroup}}
|-
| {{booklink-defined-tabular|DummitFoote|105||formal definition}} ||
|-
| {{booklink-defined-tabular|Herstein|116||formal definition, introduced between exercises}} ||
|-
| {{booklink-defined-tabular|Lang|18||definition in paragraph}} ||
|-
| {{booklink-defined-tabular|RobinsonGT|121||formal definition}} ||
|-
| {{booklink-defined-tabular|AlperinBell|95||definition in paragraph}} ||
|-
| {{booklink-defined-tabular|RobinsonAA|171||definition in paragraph}} ||
|-
| {{booklink-defined-tabular|Fraleigh|194|Definition 3.4.16|formal definition}} ||
|-
| {{booklink-defined-tabular|Hungerford|102|Definition 7.9|formal definition}} ||
|-
| {{booklink-defined-tabular|Gallian|563||}} ||
|-
| {{booklink-defined-tabular|Herstein|116||formal definition, introduced between exercises}} ||
|}

Latest revision as of 07:16, 3 August 2024

Definition

Solvable is also called soluble by some people.

Equivalent definitions in tabular format

No. Shorthand A group is termed solvable if ... A group is termed solvable if ...
1 normal series, abelian quotients there is a normal series of finite length starting from the trivial subgroup and ending at the whole group with each successive quotient group being an abelian group. there exists a series of subgroups such that each is normal in and each is abelian.
2 subnormal series, abelian quotients there is a subnormal series of finite length starting from the trivial subgroup and ending at the whole group with each successive quotient group being an abelian group. there exists a series of subgroups: such that each is normal in and each is abelian.
3 derived series finite length the derived series reaches the identity in finitely many steps the derived series of , i.e., the series where and is the derived subgroup of its predecessor, reaches the trivial subgroup in finitely many steps.
4 characteristic series, abelian quotients there is a characteristic series of finite length starting from the trivial subgroup and ending at the whole group with each successive quotient group being an abelian group. there exists a series of subgroups such that each is characteristic in and each is abelian.
5 fully invariant series, abelian quotients there is a fully invariant series of finite length starting from the trivial subgroup and ending at the whole group with each successive quotient group being an abelian group. there exists a series of subgroups such that each is fully invariant in and each is abelian.

The length of the derived series, and the smallest possible length of a series for any of the other equivalent definitions, is termed the derived length or solvable length of the group.

This definition is presented using a tabular format. |View all pages with definitions in tabular format

Equivalence of definitions

Further information: Equivalence of definitions of solvable group, equivalence of definitions of derived length

Examples

VIEW: groups satisfying this property | groups dissatisfying this property
VIEW: Related group property satisfactions | Related group property dissatisfactions

Extreme examples

Groups satisfying the property

Here are some basic/important groups satisfying the property:

 GAP ID
Cyclic group:Z22 (1)
Cyclic group:Z33 (1)
Symmetric group:S36 (1)

Here are some relatively less basic/important groups satisfying the property:

 GAP ID
Alternating group:A412 (3)
Dihedral group:D88 (3)
Quaternion group8 (4)

Here are some even more complicated/less basic groups satisfying the property:


Groups dissatisfying the property

Here are some basic/important groups that do not satisfy the property:


Here are some relatively less basic/important groups that do not satisfy the property:

 GAP ID
Alternating group:A6360 (118)
Projective special linear group:PSL(3,2)168 (42)

Here are some even more complicated/less basic groups that do not satisfy the property:



This article is about a standard (though not very rudimentary) definition in group theory. The article text may, however, contain more than just the basic definition
VIEW: Definitions built on this | Facts about this: (facts closely related to Solvable group, all facts related to Solvable group) |Survey articles about this | Survey articles about definitions built on this
VIEW RELATED: Analogues of this | Variations of this | Opposites of this |
View a complete list of semi-basic definitions on this wiki

This article defines a group property that is pivotal (i.e., important) among existing group properties
View a list of pivotal group properties | View a complete list of group properties [SHOW MORE]

The version of this for finite groups is at: finite solvable group

Metaproperties

Metaproperty name Satisfied? Proof Statement with symbols
pseudovarietal group property Yes solvability is pseudovarietal Solvability is closed under taking subgroups, quotients, and finite direct products (more below).
extension-closed group property Yes solvability is extension-closed Suppose is a normal subgroup of such that both and the quotient group are solvable groups. Then is a solvable group.
subgroup-closed group property Yes solvability is subgroup-closed If is solvable, and is a subgroup, then is solvable.
quotient-closed group property Yes solvability is quotient-closed If is solvable, and is a normal subgroup of , the quotient group is solvable.
finite direct product-closed group property Yes solvability is finite direct product-closed If are solvable, the external direct product is also solvable.
finite normal join-closed group property Yes solvability is finite normal join-closed If is a group and are all solvable normal subgroups of , the join of subgroups (in this case also the product of subgroups) is also solvable.
isoclinism-invariant group property Yes isoclinic groups have same derived length If and are isoclinic groups, then is solvable if and only if is. Moreover, if so, the derived length of equals the derived length of , unless one of the groups is trivial and the other is nontrivial abelian.

Relation with other properties

Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions Comparison
abelian group derived subgroup is trivial abelian implies solvable solvable not implies abelian (see also list of examples) Nilpotent group|FULL LIST, MORE INFO
cyclic group (see also list of examples) Abelian group, Nilpotent group|FULL LIST, MORE INFO
nilpotent group lower central series reaches the identity nilpotent implies solvable solvable not implies nilpotent (see also list of examples) |FULL LIST, MORE INFO nilpotent versus solvable
metabelian group abelian normal subgroup with abelian quotient; derived length two (see also list of examples) |FULL LIST, MORE INFO
supersolvable group normal series with cyclic factor groups supersolvable implies solvable solvable not implies supersolvable (see also list of examples) |FULL LIST, MORE INFO
polycyclic group subnormal series with cyclic factor groups
equivalent to solvable in the finite case		 polycyclic implies solvable solvable not implies polycyclic (see also list of examples) |FULL LIST, MORE INFO
metacyclic group cyclic normal subgroup with cyclic quotient group (see also list of examples) For intermediate notions between solvable group and metacyclic group, click here.

Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions Comparison
hypoabelian group transfinite derived series reaches identity;
equivalent to solvable in the finite case		 solvable implies hypoabelian hypoabelian not implies solvable |FULL LIST, MORE INFO
imperfect group no nontrivial perfect quotient group solvable implies imperfect imperfect not implies solvable |FULL LIST, MORE INFO
locally solvable group every finitely generated subgroup is solvable
equivalent to solvable in the finite case
residually solvable group every non-identity element has a non-identity image in some solvable quotient
equivalent to solvable in the finite case

Conjunction with other properties

Conjunction Other component of conjunction Additional comments
finite solvable group finite group For finite groups, being solvable is equivalent to being polycyclic, and has many other alternative characterizations.
solvable T-group T-group
solvable HN-group HN-group

Formalisms

In terms of the group extension operator

This group property can be expressed in terms of the group extension operator and/or group property modifiers that arise from this operator The group property of being solvable can be obtained in either of these equivalent ways:

Note that all these three operators have the same effect in the case of abelian groups, though in general they may not have.

Testing

The testing problem

Further information: Solvability testing problem

The problem of testing whether a group is solvable or not reduces to the problem of computing its derived series. This can be done when the group is described by means of a generating set, if the normal closure algorithm can be implemented.

GAP command

This group property can be tested using built-in functionality of Groups, Algorithms, Programming (GAP).
The GAP command for this group property is:IsSolvableGroup
View GAP-testable group properties

To determine whether a group is solvable or not, we cna use the following GAP command: 

IsSolvableGroup(group);

where group may be a definition of the group or a name for a group previously defined.

Study of this notion

Mathematical subject classification

Under the Mathematical subject classification, the study of this notion comes under the class: 20F16

The class 20F16 is used for the general theory of solvable groups, while the class 20D10 (coming under 20D which is for finite groups) focusses on finite solvable groups.

Also closely related is 20F19: Generalizations of nilpotent and solvable groups.

References

Textbook references

Book Page number Chapter and section Contextual information View
Abstract Algebra by David S. Dummit and Richard M. Foote, 10-digit ISBN 0471433349, 13-digit ISBN 978-0471433347More info 105 formal definition
Topics in Algebra by I. N. HersteinMore info 116 formal definition, introduced between exercises
Algebra by Serge Lang, ISBN 038795385XMore info 18 definition in paragraph
A Course in the Theory of Groups by Derek J. S. Robinson, ISBN 0387944613More info 121 formal definition
Groups and representations by Jonathan Lazare Alperin and Rowen B. Bell, ISBN 0387945261More info 95 definition in paragraph
An Introduction to Abstract Algebra by Derek J. S. Robinson, ISBN 3110175444More info 171 definition in paragraph
A First Course in Abstract Algebra (6th Edition) by John B. Fraleigh, ISBN 0201763907More info 194 Definition 3.4.16 formal definition
Algebra (Graduate Texts in Mathematics) by Thomas W. Hungerford, ISBN 0387905189More info 102 Definition 7.9 formal definition
Contemporary Abstract Algeba by Joseph Gallian, ISBN 0618514716More info 563
Topics in Algebra by I. N. HersteinMore info 116 formal definition, introduced between exercises
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