Equivalence of definitions of solvable group, equivalence of definitions ... finite solvable group
==Metaproperties==
{| class="sortable" ... ...
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#REDIRECT Group whose automorphism group is solvable ... ...
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| Weaker than::solvable group || || || ||
|-
| Weaker than::residually nilpotent group || || || ||
|}
===Weaker properties===
{| class="sortable ... ...
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A group is a virtually solvable group if it has a subgroup of finite index that is solvable.
==Relation with other properties==
===Stronger properties ... ...
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#redirect polycyclic group ... ...
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#redirect polycyclic group ... ...
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if every finitely generated subgroup of it is a solvable group. ... locally operator|solvable group
==Relation with other properties== ... ...
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finite group. Then, G is termed a \pi-solvable group if G is a p-solvable ... * Weaker than::Solvable group
===Weaker properties===
* Stronger than::Pi ... ...
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a prime number. We say that G is a p-solvable group if it satisfies the following ... Further, a finite group is a finite solvable group if it is p-solvable for ... ...
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Note that the notion of n-solvable group is distinct from the notion of p-solvable group. In particular, when somebody talks of n-solvability ... ...
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A group is termed a periodic solvable group or locally finite solvable ... generated subgroup is finite) and a solvable group (the derived series reaches ... ...
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A complete solvable group is a group that satisfies both these conditions: ... * It is a solvable group.
==Relation with other properties== ... ...
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==Definition==
A group is said to be splitting-solvable if it has a splitting series with all the quotient groups Abelian. ... ...
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A defining ingredient::finite group is termed a finite solvable group ... # It is a defining ingredient::solvable group
# It is a defining ingredient: ... ...
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#redirect periodic solvable group ... ...
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A finitely presented solvable group is a group that is both a finitely ... finitely presented group|solvable group
==Relation with other properties== ... ...
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subgroup|solvable group
==Definition== A subgroup H of a group G is termed a subgroup of solvable group if ... ...
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finitely generated group|solvable group
==Definition== A finitely generated solvable group is a group that is both a finitely ... ...
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==Definition==
===Symbol-free definition===
A family of finite groups is said to be smoothly solvable if all its members are solvable and further ... ...
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* G is a defining ingredient::p-solvable group.
* Either p \ge 5 or ... * Stronger than::p-solvable group: strongly p-solvable implies p-solvable ... ...
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==Definition==
A solvable variety of groups or soluble variety of groups is a subvariety of the variety of groups (i.e., a collection of groups ... ...
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# Every group in the variety is a solvable group
# There exists a nonnegative integer l such that every group in the variety is solvable with ... ...
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* G is a defining ingredient::p-solvable group.
* Either p \ge 5 or ... * Stronger than::p-solvable group: strongly p-solvable implies p-solvable ... ...
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A complete solvable group is a group that satisfies both these conditions: ... * It is a solvable group.
==Relation with other properties== ... ...
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==Statement==
If a finite group is a monomial group (sometimes referred to as an M-group or M_1-group), it is solvable. This result goes by the ... ...
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weaker = solvable group}}
==Statement==
Suppose G is an group whose automorphism group is solvable, i.e., a group whose automorphism group is ... ...
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stronger = solvable group|
weaker = group whose automorphism group ... It is possible to have a solvable group G such that the automorphism ... ...
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if every finitely generated subgroup of it is a solvable group. ... locally operator|solvable group
==Relation with other properties== ... ...
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Given: A finite non-solvable group G.
To prove: There exists a minimal simple group arising as a subquotient of G.
Proof: We prove this by induction ... ...
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old property = solvable group|
new generic context = IAPS| ... * Every member is a solvable group, and there is a common finite bound ... ...
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An automorphism of a solvable group is said to be solvable-extensible ... Let G be a solvable group and \sigma an automorphism of G. Then \sigma ... ...
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finitely generated group|solvable group
==Definition== A finitely generated solvable group is a group that is both a finitely ... ...
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group if its automorphism group is a solvable group.
==Relation with ... | Weaker than::complete solvable group || solvable group that is also ... ...
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Suppose p is an odd prime and G is a strongly p-solvable group. Then ... In particular, for p \ge 5, any fact about::p-solvable group is of ... ...
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characteristic subgroup|solvable group
==Definition==
A subgroup of a group is termed a solvable characteristic subgroup if it is solvable as ... ...
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A finitely presented solvable group is a group that is both a finitely ... finitely presented group|solvable group
==Relation with other properties== ... ...
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A solvable group generated by finitely many periodic elements is a ... * periodic solvable group: Clearly, being periodic and solvable does ... ...
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Let G be a solvable group. Then, the following are equivalent for G: * G is a solvable group (in particular, it is a finite solvable group). ... ...
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characteristic subgroup|solvable group
==Definition== is termed a characteristic subgroup of solvable group if G is a solvable group ... ...
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* G_1 is a solvable group if and only if G_2 is a solvable group. ... G_1 is trivial). || definition of solvable group, via the derived series. || -- ... ...
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