# Superperfect group

This page describes a group property obtained as a conjunction (AND) of two (or more) more fundamental group properties: perfect group and Schur-trivial group
View other group property conjunctions OR view all group properties

## Definition

### Equivalent definitions in tabular format

No. Shorthand A group is termed superperfect if ... A group $G$ is termed superperfect if ...
1 first two homology groups vanish its first two homology groups for trivial group action, with coefficients in the integers, vanish. $H_1(G;\mathbb{Z})$ and $H_2(G;\mathbb{Z})$ are both trivial groups.
2 perfect and Schur-trivial it is a perfect group (this corresponds to the first homology group, i.e., the abelianization, vanishing) as well as a Schur-trivial group (this corresponds to the second homology group, i.e., the Schur multiplier, vanishing) the abelianization $G^{\operatorname{ab}} = G/[G,G] = H_1(G;\mathbb{Z})$ is trivial and the Schur multiplier $M(G) = H_2(G;\mathbb{Z})$ is trivial.
3 perfect and all second cohomology groups for trivial action vanish it is a perfect group and its second cohomology group for trivial group action with respect to every possible abelian group is trivial. $G$ is perfect and $H^2(G;A)$ (with the trivial group action) is trivial for every abelian group $A$.
4 perfect and every central extension is split it is a perfect group and every central extension with it as the quotient group splits. $G$ is perfect and for any group $E$ with a central subgroup $A$ such that $E/A \cong G$, $A$ is a direct factor of $E$, i.e., the extension splits.
5 commutator map from exterior square is isomorphism to whole group the commutator map homomorphism from the exterior square is an isomorphism to the whole group (the surjectivity corresponds to the group being perfect, because the abelianization is the cokernel, and the injectivity corresponds to the group being Schur-trivial, because the Schur multiplier is the kernel). the natural commutator map $G \wedge G \to G$ is an isomorphism.
6 commutator map from tensor square is isomorphism to whole group the commutator map, viewed as a homomorphism from the tensor square (via first mapping to the exterior square, then applying the commutator) is an isomorphism to the whole group. the natural commutator map $G \otimes G \to G$ is an isomorphism.
This definition is presented using a tabular format. |View all pages with definitions in tabular format

## Examples

### Groups satisfying the property

Here are some basic/important groups satisfying the property:

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

GAP ID
Special linear group:SL(2,5)120 (5)

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:A560 (5)
Dihedral group:D88 (3)
Symmetric group:S5120 (34)

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

## Metaproperties

Metaproperty name Satisfied? Proof Statement with symbols
subgroup-closed group property No superperfectness is not subgroup-closed It is possible to have a superperfect group $G$ and a subgroup $H$ of $G$ such that $H$ is not superperfect.
quotient-closed group property No superperfectness is not quotient-closed It is possible to have a superperfect group $G$ and a normal subgroup $H$ of $G$ such that the quotient group $G/H$ is not superperfect.
finite direct product-closed group property Yes superperfectness is finite direct product-closed If $G_1,G_2,\dots,G_n$ are (possibly isomorphic, possibly non-isomorphic) groups, each of which is superperfect, then the external direct product $G_1 \times G_2 \times \dots \times G_n$ is also superperfec.t

## Relation with other properties

### Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
perfect group it equals its own derived subgroup |FULL LIST, MORE INFO
Schur-trivial group its Schur multiplier is trivial follows from Schur multiplier of cyclic group is trivial |FULL LIST, MORE INFO