Second cohomology group for trivial group action of UT(3,p) on Zp

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Description of the group

This article discusses the second cohomology group for trivial group action:

\! H^2(G;A)

where \! G = UT(3,p) is unitriangular matrix group:UT(3,p), i.e., it is the unitriangular matrix group of degree three over the field of p elements and A = \mathbb{Z}_p is the group of prime order.

For p \ge 5, the group is isomorphic to an elementary abelian group of prime-fourth order.

The cases p = 2 and p = 3 differ somewhat from the general case. See:

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.

As noted above, some of the details differ for the p = 2 and p = 3 cases.

Cohomology class type Number of cohomology classes Corresponding group extension GAP ID 2nd part (GAP ID is (p^4,2nd part) Stem extension Base characteristic in whole group? Nilpotency class of whole group (at least 2, at most 3) Derived length of whole group (always exactly 2) Minimum size of generating set of whole group (at least 2, at most 3)
trivial 1 direct product of U(3,p) and Zp 12 No No 2 2 3
symmetric and nontrivial p^2 - 1 SmallGroup(p^4,3) 3 No Yes 2 2 2
cyclicity-preserving and nontrivial p^2 - 1 SmallGroup(p^4,7) 7 Yes Yes 3 2 2
nontrivial (p^2 - 1)(p - 1) SmallGroup(p^4,8) 8 Yes Yes 3 2 2
nontrivial (p^2 - 1)(p^2 - p)/2 SmallGroup(p^4,9) 9 Yes Yes 3 2 2
nontrivial (p^2 - 1)(p^2 - p)/2 SmallGroup(p^4,10) 10 Yes Yes 3 2 2
Total (6 rows) p^4 (equals order of the cohomology group) -- -- -- -- -- -- --

Note that all these extensions are central extensions with the base normal subgroup isomorphic to group of prime order and the quotient group isomorphic to prime-cube order group:U(3,p). Due to the fact that order of extension group is product of order of normal subgroup and quotient group, the order of each of the extension groups is p \times p^3 = p^4.

Some, but not all, of the extensions are stem extensions. These are not the Schur covering groups, but rather are quotients of the Schur covering group, which has order p^5.

The minimum size of generating set of the extension group is at least equal to 2 (which is the minimum size of generating set of the quotient group) and at most equal to 3 (which is the sum of the minimum size of generating set of the normal subgroup and the quotient group).

The nilpotency class is at least 2 and at most 3 in all cases. It is at least 2 because the quotient prime-cube order group:U(3,p) has nilpotency class two. It is at most 3 because the sum of the nilpotency class of the normal subgroup and quotient group is 3, and the extension is a central extension. The derived length is always exactly 2 because nilpotency class 2 or 3 forces derived length exactly 2, using derived length is logarithmically bounded by nilpotency class.

Group actions

Under the action of the automorphism group of the acting group

Each of the cohomology class types mentioned above forms one orbit under the action of the automorphism group of U(3,p).

Subgroups of interest

Subgroup Value Corresponding group extensions GAP IDs second part Group extension groupings for each coset GAP IDs second part
subgroup generated by images of symmetric 2-cocycles, equivalently, image of \operatorname{Ext}^1(G^{\operatorname{ab}},A) in the direct sum decomposition arising from the universal coefficients theorem (this means that the restriction to the derived subgroup of the acting group gives a split extension and all the interesting stuff is happening at the level of the abelianization of the acting group) elementary abelian group of prime-square order direct product of U(3,p) and Zp (1 time), SmallGroup(p^4,3) (p^2 - 1 times) 12,3 (direct product of U(3,p) and Zp, SmallGroup(p^4,3) (p^2 - 1 times)) (1 copy), (SmallGroup(p^4,7) (1 time), SmallGroup(p^4,8) (p - 1 times), SmallGroup(p^4,9) ((p^2-p)/2 times), SmallGroup(p^4,10) ((p^2 - p)/2 times)) (p^2 - 1 copies) (12,3) (1 copy), (7,8,9,10) (p^2 - 1 copies)
subgroup generated by images of cyclicity-preserving 2-cocycles elementary abelian group of prime-square order direct product of U(3,p) and Zp (1 time), SmallGroup(p^4,7) (p^2 - 1 times) 12,7 (direct product of U(3,p) and Zp (1 time), SmallGroup(p^4,7) (p^2 - 1 times)) (1 copy), (SmallGroup(p^4,3) (1 time), SmallGroup(p^4,8) (p - 1 times), SmallGroup(p^4,9) ((p^2-p)/2 times), SmallGroup(p^4,10) ((p^2 - p)/2 times)) (p^2 - 1 copies) (12,7) (1 copy), (3,8,9,10) (p^2 - 1 copies)

Direct sum decomposition

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

General background

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

0 \to \operatorname{Ext}^1_{\mathbb{Z}}(G^{\operatorname{ab}},A) \to H^2(G;A) \stackrel{\operatorname{Skew}}{\to} \operatorname{Hom}(H_2(G;\mathbb{Z}),A) \to 0

where G^{\operatorname{ab}} is the abelianization of G and its image comprises those extensions where the restricted extension of the derived subgroup [G,G] on A is trivial and the corresponding extension of the quotient group is abelian. Also, H_2(G;\mathbb{Z}) is the Schur multiplier of G.

We also know, again from the general theory, that the short exact sequence above splits, i.e., the image of \operatorname{Ext}^1_{\mathbb{Z}}(G^{\operatorname{ab}},A) in H^2(G;A) has a complement inside H^2(G;A). However, there need not in general be a natural or even an automorphism-invariant choice of splitting.

In this case

For this choice of G and A, G^{\operatorname{ab}} is isomorphic to the elementary abelian group of prime-square order. The corresponding group \operatorname{Ext}^1_{\mathbb{Z}}(G^{\operatorname{ab}},A) is also isomorphic to the elementary abelian group of prime-square order.

The Schur multiplier H_2(G;\mathbb{Z}) is the elementary abelian group of prime-square order, hence \operatorname{Hom}(H_2(G;\mathbb{Z}),A) is also isomorphic to elementary abelian group of prime-square order.

The image of \operatorname{Ext}^1_{\mathbb{Z}}(G^{\operatorname{ab}},A) in H^2(G;A) comprises the p^2 non-stem extensions, which give groups of overall nilpotency class two. It has p^2 cosets in the whole cohomology group, and all the p^2 - 1 non-identity cosets are stem extensions giving groups of overall nilpotency class three.

To split the short exact sequence in an automorphism-invariant fashion, we could pick as a complement the subgroup generated by images of cyclicity-preserving 2-cocycles: comprising the identity element (extension direct product of U(3,p) and Zp) and the p^2 - 1 cyclicity-preserving extensions that give SmallGroup(p^4,7).

A quick schematic of the picture is:

direct product of U(3,p) and Zp SmallGroup(p^4,3) (p^2 - 1 times)
SmallGroup(p^4,7) (p^2 - 1 times) stuff here includes SmallGroup(p^4,8), SmallGroup(p^4,9), SmallGroup(p^4,10)

The actual picture has p^2 - 1 distinct columns in place of the second column and p^2 - 1 distinct rows in place of the second row. Each row except the first contains, in its columns other than the first column, p - 1 copies of SmallGroup(p^4,8) and (p^2 - p)/2 copies each of SmallGroup(p^4,9) and SmallGroup(p^4,10). Each column except the first contains, in its rows other than the first row, p - 1 copies of SmallGroup(p^4,8) and (p^2 - p)/2 copies each of SmallGroup(p^4,9) and SmallGroup(p^4,10).