M16: Difference between revisions

Revision as of 01:05, 9 August 2010

Definition

The group, sometimes denoted $M_{16}$ , is defined as follows:

$M_{16} = ⟨ a, x ∣ a^{8} = x^{2} = e, x a x = a^{5} ⟩$ .

Here, $e$ denotes the identity element.

Arithmetic functions

Want to compare and contrast arithmetic function values with other groups of the same order? Check out groups of order 16#Arithmetic functions

Function	Value	Similar groups	Explanation for function value
underlying prime of p-group	2
order (number of elements, equivalently, cardinality or size of underlying set)	16	groups with same order
prime-base logarithm of order	4	groups with same prime-base logarithm of order
max-length of a group	4		max-length of a group equals prime-base logarithm of order for group of prime power order
chief length	4		chief length equals prime-base logarithm of order for group of prime power order
composition length	4		composition length equals prime-base logarithm of order for group of prime power order
exponent of a group	8	groups with same order and exponent of a group \| groups with same prime-base logarithm of order and exponent of a group \| groups with same exponent of a group
prime-base logarithm of exponent	3	groups with same order and prime-base logarithm of exponent \| groups with same prime-base logarithm of order and prime-base logarithm of exponent \| groups with same prime-base logarithm of exponent
Frattini length	3	groups with same order and Frattini length \| groups with same prime-base logarithm of order and Frattini length \| groups with same Frattini length
nilpotency class	2	groups with same order and nilpotency class \| groups with same prime-base logarithm of order and nilpotency class \| groups with same nilpotency class
derived length	2	groups with same order and derived length \| groups with same prime-base logarithm of order and derived length \| groups with same derived length
minimum size of generating set	2	groups with same order and minimum size of generating set \| groups with same prime-base logarithm of order and minimum size of generating set \| groups with same minimum size of generating set
subgroup rank of a group	2	groups with same order and subgroup rank of a group \| groups with same prime-base logarithm of order and subgroup rank of a group \| groups with same subgroup rank of a group
rank of a p-group	2	groups with same order and rank of a p-group \| groups with same prime-base logarithm of order and rank of a p-group \| groups with same rank of a p-group
normal rank of a p-group	2	groups with same order and normal rank of a p-group \| groups with same prime-base logarithm of order and normal rank of a p-group \| groups with same normal rank of a p-group
characteristic rank of a p-group	2	groups with same order and characteristic rank of a p-group \| groups with same prime-base logarithm of order and characteristic rank of a p-group \| groups with same characteristic rank of a p-group

Group properties

Property	Satisfied	Explanation
Abelian group	No	$a, x$ do not commute
Nilpotent group	Yes	prime power order implies nilpotent
Metacyclic group	Yes
Supersolvable group	Yes
Solvable group	Yes

Subgroup structure

Further information: subgroup structure of M16

To describe subgroups, we use the defining presentation given at the beginning:

$M_{16} = ⟨ a, x ∣ a^{8} = x^{2} = e, x a x = a^{5} ⟩$ .

The subgroups are as follows:

The trivial subgroup. Isomorphic to trivial group. (1)
The two-element subgroup $⟨ a^{4} ⟩$ . This is the derived subgroup, and is also the socle. In particular, it is a characteristic subgroup. Isomorphic to cyclic group:Z2. (1)
The two-element subgroups $⟨ x ⟩$ and $⟨ a^{4} x ⟩$ . These are conjugate subgroups. Isomorphic to cyclic group:Z2. (2)
The four-element subgroup $⟨ a^{2} ⟩$ . This is the center, and is also the Frattini subgroup. In particular, it is a characteristic subgroup. Isomorphic to cyclic group:Z4. (1)
The four-element subgroup $⟨ a^{2} x ⟩$ . This is a characteristic subgroup. Isomorphic to cyclic group:Z4. (1)
The four-element subgroup $⟨ a^{4}, x ⟩$ . This is a characteristic subgroup. Isomorphic to Klein four-group. (1)
The eight-element subgroups $⟨ a ⟩$ and $⟨ a x ⟩$ . These are both normal subgroups and are automorphic subgroups -- an outer automorphism interchanges them. Isomorphic to cyclic group:Z8. (2)
The eight-element subgroup $⟨ a^{2}, x ⟩$ . This is a characteristic subgroup. Isomorphic to direct product of Z4 and Z2. (1)
The whole group. (1)

GAP implementation

Group ID

This finite group has order 16 and has ID 6 among the groups of order 16 in GAP's SmallGroup library. For context, there are groups of order 16. It can thus be defined using GAP's SmallGroup function as:

SmallGroup(16,6)

For instance, we can use the following assignment in GAP to create the group and name it $G$ :

gap> G := SmallGroup(16,6);

Conversely, to check whether a given group $G$ is in fact the group we want, we can use GAP's IdGroup function:

IdGroup(G) = [16,6]

or just do:

IdGroup(G)

to have GAP output the group ID, that we can then compare to what we want.

@@ Line 55: / Line 55: @@
 |[[Satisfies property::Solvable group]] || Yes || ||
 |}
+==Subgroup structure==
+{{further|[[subgroup structure of M16]]}}
+To describe subgroups, we use the defining presentation given at the beginning:
+<math>M_{16} = \langle a,x \mid a^8 = x^2 = e, xax = a^5 \rangle</math>.
+The subgroups are as follows:
+# The trivial subgroup. Isomorphic to [[subgroup::trivial group]]. (1)
+# The two-element subgroup <math>\langle a^4 \rangle</math>. This is the [[derived subgroup]], and is also the [[socle]]. In particular, it is a [[characteristic subgroup]]. Isomorphic to [[subgroup::cyclic group:Z2]]. (1)
+# The two-element subgroups <math>\langle x \rangle</math> and <math>\langle a^4x \rangle</math>. These are [[conjugate subgroups]]. Isomorphic to [[subgroup::cyclic group:Z2]]. (2)
+# The four-element subgroup <math>\langle a^2 \rangle</math>. This is the [[center]], and is also the [[Frattini subgroup]]. In particular, it is a [[characteristic subgroup]]. Isomorphic to [[subgroup::cyclic group:Z4]]. (1)
+# The four-element subgroup <math>\langle a^2x \rangle</math>. This is a [[characteristic subgroup]]. Isomorphic to [[subgroup::cyclic group:Z4]]. (1)
+# The four-element subgroup <math>\langle a^4, x \rangle</math>. This is a [[characteristic subgroup]]. Isomorphic to [[subgroup::Klein four-group]]. (1)
+# The eight-element subgroups <math>\langle a \rangle</math> and <math>\langle ax \rangle</math>. These are both [[normal subgroup]]s and are [[automorphic subgroups]] -- an outer automorphism interchanges them. Isomorphic to [[subgroup::cyclic group:Z8]]. (2)
+# The eight-element subgroup <math>\langle a^2,x \rangle</math>. This is a [[characteristic subgroup]]. Isomorphic to [[subgroup::direct product of Z4 and Z2]]. (1)
+# The whole group. (1)
 ==GAP implementation==
 {{GAP ID|16|6}}