Second cohomology group for trivial group action of E8 on Z4

This article gives information about the second cohomology group for trivial group action (i.e., the second cohomology group with trivial action) of the group elementary abelian group:E8 on cyclic group:Z4. The elements of this classify the group extensions with cyclic group:Z4 in the center and elementary abelian group:E8 the corresponding quotient group. Specifically, these are precisely the central extensions with the given base group and acting group.
The value of this cohomology group is elementary abelian group:E64.
Get more specific information about elementary abelian group:E8 |Get more specific information about cyclic group:Z4|View other constructions whose value is elementary abelian group:E64

Description of the group

We consider here the second cohomology group for trivial group action of elementary abelian group:E8 on the cyclic group:Z4, i.e.,

$\!H^{2}(G,A)$

where $G\cong E_{8}\cong \mathbb {Z} _{2}\times \mathbb {Z} _{2}\times \mathbb {Z} _{2}$ and $A\cong \mathbb {Z} _{4}$.

The cohomology group is isomorphic to elementary abelian group:E64.

Elements

FACTS TO CHECK AGAINST (second cohomology group for trivial group action):
Background reading on relationship with extension groups: Group extension problem
Arithmetic functions of extension group:
order (thus all extension groups have the same order): order of extension group is product of order of normal subgroup and quotient group
nilpotency class: nilpotency class of extension group is between nilpotency class of quotient group and one more for central extension
derived length: derived length of extension group is bounded by sum of derived length of normal subgroup and quotient group
minimum size of generating set: minimum size of generating set of extension group is bounded by sum of minimum size of generating set of normal subgroup and quotient group|minimum size of generating set of quotient group is at most minimum size of generating set of group

WHAT'S THE TABLE BELOW?: Recall that there is a correspondence:
Elements of the group $H^{2}(G;A)$ for the trivial group action $\leftrightarrow$ congruence classes of central extensions with the specified subgroup $A$ and quotient group $G$.
This descends to a correspondence:
Orbits for the group action of $\operatorname {Aut} (G)\times \operatorname {Aut} (A)$ on $H^{2}(G;A)$ $\leftrightarrow$ pseudo-congruence classes of central extensions.
The table below breaks down the second cohomology group as a union of these orbits, with (as a general rule) each row describing one orbit, i.e., one "cohomology class type", aka one "pseudo-congruence class" of central extensions. The number of rows is the number of pseudo-congruence classes of central extensions.

Cohomology class type	Number of cohomology classes	Corresponding group extension	GAP ID (second part, order is 32)	Stem extension?	Base characteristic in whole group?	Hall-Senior family (equivalence class up to being isoclinic)	Nilpotency class of whole group (at least 1, at most 2)	Derived length of whole group (at least 1, at most 2)	Minimum size of generating set of whole group (at least 3, at most 4)	Subgroup information on base in whole group
trivial	1	direct product of E8 and Z4	45	No	No	$\Gamma _{1}$	1	1	4
symmetric and nontrivial	7	direct product of Z8 and V4	36	No	No	$\Gamma _{1}$	1	1	3
one of the non-symmetric ones	28	central product of D8 and Z8	38	No	Yes	$\Gamma _{2}$	2	2	3
one of the non-symmetric ones	21	direct product of M16 and Z2	37	No		$\Gamma _{2}$	2	2	3
one of the non-symmetric ones	7	direct product of SmallGroup(16,13) and Z2	48	No		$\Gamma _{2}$	2	2	4
Total (5 rows)	64	--	--	--	--	--	--	--	--

Group actions

Under the action of the automorphism group of the acting group

The acting group $G$, which is isomorphic to elementary abelian group:E8, has a huge automorphism group, general linear group:GL(3,2) of order $168$. Each cohomology class type in the table above is one orbit. In particular, the trivial group extension is the only fixed point, there are two orbits of size $7$, and there is one orbit each of size $21$ and $28$.

Subgroups of interest

Subgroup	Value	Corresponding group extensions for subgroup	GAP IDs second part	Group extension groupings for each coset	GAP IDs second part
IIP subgroup of second cohomology group for trivial group action	elementary abelian group:E8	direct product of E8 and Z4, direct product of SmallGroup(16,13) and Z2	45, 48	(direct product of E8 and Z4, direct product of SmallGroup(16,13) and Z2 (7 copies)) (1 time), (direct product of Z8 and V4, direct product of M16 and Z2 (3 times), central product of D8 and Z8 (4 times)) (7 copies)	(45, 48), (36,37,38)
cyclicity-preserving subgroup of second cohomology group for trivial group action	elementary abelian group:E8	direct product of E8 and Z4, direct product of SmallGroup(16,13) and Z2	45, 48	(direct product of E8 and Z4, direct product of SmallGroup(16,13) and Z2 (7 copies)) (1 time), (direct product of Z8 and V4, direct product of M16 and Z2 (3 times), central product of D8 and Z8 (4 times)) (7 copies)	(45, 48), (36,37,38)
subgroup generated by images of symmetric 2-cocycles (corresponds to abelian group extensions)	elementary abelian group:E8	direct product of E8 and Z4, direct product of Z8 and V4 (7 times)	45, 36	(direct product of E8 and Z4, direct product of Z8 and V4 (7 copies)) (1 time), (direct product of SmallGroup(16,13) and Z2, direct product of M16 and Z2 (3 copies), central product of D8 and Z8 (4 copies)) (7 times)	(45, 36), (48, 37, 38)

Direct sum decomposition

For background information, see formula for second cohomology group for trivial group action of abelian group in terms of Schur multiplier and abelianization

General background

We know from the general theory that there is a natural short exact sequence:

$0\to \operatorname {Ext} _{\mathbb {Z} }^{1}(G,A)\to H^{2}(G;A){\stackrel {\operatorname {Skew} }{\to }}\operatorname {Hom} (\bigwedge ^{2}G,A)\to 0$

where the image of $\operatorname {Ext} ^{1}$ is $H_{sym}^{2}(G;A)$, i.e., the group of cohomology classes represented by symmetric 2-cocycles. We also know, again from the general theory, that the short exact sequence above splits, i.e., $H_{sym}^{2}(G;A)$ has a complement inside $H^{2}$. However, there need not in general be a natural or even an automorphism-invariant choice of splitting.

In this case

See also second cohomology group for trivial group action is internal direct sum of symmetric and cyclicity-preserving 2-cocycle subgroups if acting group is elementary abelian 2-group and every element of order two in the base group is a square

In terms of the general background, one way of putting this is that the skew map:

$H^{2}(G;A){\stackrel {\operatorname {Skew} }{\to }}\operatorname {Hom} (\bigwedge ^{2}G,A)$

has a section (i.e., a reverse map):

$\operatorname {Hom} (\bigwedge ^{2}G,A)\to H^{2}(G;A)$

whose image is $H_{CP}^{2}(G;A)$ of cohomology classes represented by cyclicity-preserving 2-cocycles (see cyclicity-preserving subgroup of second cohomology group for trivial group action). Thus, the natural short exact sequence splits, and we get an internal direct sum decomposition:

$H^{2}(G;A)=H_{sym}^{2}(G;A)+H_{CP}^{2}(G;A)$

A pictorial description of this is as follows. Here, each column is a coset of $H_{CP}^{2}(G,A)$ and each row is a coset of $H_{sym}^{2}(G,A)$. The top left entry is the identity element, hence the top row corresponds to abelian group extensions and the left column corresponds to cyclicity-preserving 2-cocycles.

Note that apart from the choice of first row and first column representing the subgroups, the ordering of rows and columns has no significance, since the automorphism group acts transitively on all the rows other than the first row and all the columns other than the first column. Also, note that the $7\times 7$ matrix that we get on deleting the first row and first column has the configuration of a Fano plane -- such a configuration is dictated by the various symmetry considerations.

direct product of E8 and Z4	direct product of Z8 and V4	direct product of Z8 and V4	direct product of Z8 and V4	direct product of Z8 and V4	direct product of Z8 and V4	direct product of Z8 and V4	direct product of Z8 and V4
direct product of SmallGroup(16,13) and Z2	direct product of M16 and Z2	central product of D8 and Z8	central product of D8 and Z8	central product of D8 and Z8	direct product of M16 and Z2	central product of D8 and Z8	direct product of M16 and Z2
direct product of SmallGroup(16,13) and Z2	direct product of M16 and Z2	direct product of M16 and Z2	central product of D8 and Z8	central product of D8 and Z8	central product of D8 and Z8	direct product of M16 and Z2	central product of D8 and Z8
direct product of SmallGroup(16,13) and Z2	central product of D8 and Z8	direct product of M16 and Z2	central product of D8 and Z8	central product of D8 and Z8	direct product of M16 and Z2	central product of D8 and Z8	direct product of M16 and Z2
direct product of SmallGroup(16,13) and Z2	central product of D8 and Z8	direct product of M16 and Z2	direct product of M16 and Z2	central product of D8 and Z8	direct product of M16 and Z2	central product of D8 and Z8	central product of D8 and Z8
direct product of SmallGroup(16,13) and Z2	central product of D8 and Z8	central product of D8 and Z8	central product of D8 and Z8	direct product of M16 and Z2	direct product of M16 and Z2	direct product of M16 and Z2	central product of D8 and Z8
direct product of SmallGroup(16,13) and Z2	central product of D8 and Z8	central product of D8 and Z8	direct product of M16 and Z2	central product of D8 and Z8	central product of D8 and Z8	direct product of M16 and Z2	direct product of M16 and Z2
direct product of SmallGroup(16,13) and Z2	direct product of M16 and Z2	central product of D8 and Z8	direct product of M16 and Z2	direct product of M16 and Z2	central product of D8 and Z8	central product of D8 and Z8	central product of D8 and Z8

Generalized Baer Lie rings

The examples here illustrate the cocycle halving generalization of Baer correspondence.

The direct sum decomposition (discussed in the preceding section):

$H^{2}(G;A)=H_{sym}^{2}(G;A)+H_{CP}^{2}(G;A)$

gives rise to some examples of the cocycle halving generalization of Baer correspondence. For any group extension arising as an element of $H^{2}(G;A)$, the additive group of its Lie ring arises as the group extension corresponding to the projection onto $H_{sym}^{2}(G;A)$, and the Lie bracket coincides with the group commutator.

In the description (in the preceding section), the additive group of the Lie ring of a given group is the unique abelian group in the column corresponding to that group.

Thus, we have the following correspondences emerging:

Group	GAP ID	Additive group of Lie ring	GAP ID	More about the correspondence
direct product of SmallGroup(16,13) and Z2	(32,48)	direct product of E8 and Z4	(32,45)	generalized Baer correspondence between direct product of SmallGroup(16,13) and Z2 and direct product of E8 and Z4
central product of D8 and Z8	(32,38)	direct product of Z8 and V4	(32,36)	generalized Baer correspondence between central product of D8 and Z8 and direct product of Z8 and V4
direct product of M16 and Z2	(32,37)	direct product of Z8 and V4	(32,36)	generalized Baer correspondence between direct product of M16 and Z2 and direct product of Z8 and V4