1
vote
0answers
30 views

finite group homology: $nH_k(G;M)=0$ for $n=|G|$?

Let $G$ be a finite group. Is there a simple proof (if any) that the order of $G$ annihilates the Eilenberg-MacLane homology $H_k(G;M)$ for all $k\geq1$? A simple proof of the statement for ...
4
votes
0answers
51 views

On the definition of groups of multiplicative type

Let $k$ be a field of characteristic 0. The definition of a linear algebraic $k$-group of multiplicative type (m.t.) I've seen the most in the literature is that $G$ is of m.t. if it is a ...
1
vote
1answer
40 views

First cohomology group of direct products

Let $p$ be a prime number and H be a finite group with $|H|=p-1$ and consider $\varphi: H \times Z_{p^k} \rightarrow Aut(Z_{p^k})$ as a non-trivial action of $H \times Z_{p^k}$ on $Z_{p^k}$ such that ...
4
votes
2answers
127 views

Action of $G/H$ on $H_n(H;M)$

I'm currently studying group cohomology and have trouble with the Hochschild-Serre spectral sequence. My problem is this: Given a short exact sequence of groups $$ 0 \to H \to G \to G/H \to 0$$ how ...
3
votes
2answers
83 views

The semidirect product as a deformation of the direct product

The way I think of the semidirect product is as a "deformation" of the direct product. Is there a way of making this intuition precise? Perhaps using some certain (co-) homology theory of groups?
3
votes
1answer
42 views

Computing the order of the first cohomology group $|H^1(S_n, \mathbb F_p^n)|$

Assume $n\geq 3$, $p$ is a prime, and that $S_n$ acts on $V=\mathbb F_p^n$ by permuting the basis vectors $v_1,\ldots, v_n$. I want to compute the order of the first cohomology group of this action. ...
1
vote
1answer
132 views

What can we say about groups $G$ with $H_3(G)=0$?

Let $G$ be a group. What can we say about groups such that $H_3(G)=0$? If a characterization is not possible, then knowing examples of such groups would be good? Any help is appreciated. Thanks
1
vote
0answers
52 views

Pushout and pullback of short exact sequence of groups

I think that there might be some textbooks which introduce the notions of pushout and pullback of a short exact sequence of groups. However, I cannot find any of them. To be precise, for a given ...
1
vote
1answer
72 views

Period of a particular finite group

Let $G$ be a group fitting in the following exact sequence: $0 \to \mathbb{Z}/p \to G \to \mathbb{Z}/q^r \to 0.$ Here $q$ and $p$ are primes (not necessarily distinct). It is easy to check (by the ...
2
votes
1answer
136 views

Splitting short exact sequence of space groups

I want to prove the following: Assume we have two space groups $G,G^\prime \subseteq \text{Euc}(V) \subseteq \text{Aff}(V)$ which are affinely equivalent, $G \sim G^\prime, \; \text{ i.e. }\; ...
2
votes
1answer
76 views

An exact sequence of unit groups

In the answer of K. Conrad to this question, he mentions a "nice 4-term short exact sequence of abelian groups (involving units groups mod a, mod b, and mod ab)" proving the product formula for ...
1
vote
2answers
60 views

If $\text{Ext}^1(\mathbb{Q}/ \mathbb{Z}, D ) = 0$ then $D$ is divisible

This is Exercise 7.15(ii) from Rotman's book, Introduction to homological algebra that I'm doing. If $D$ is an abelian group and $\text{Ext}^1(\mathbb{Q}/ \mathbb{Z}, D ) = 0$, prove that $D$ is ...
2
votes
1answer
58 views

$\hom_{\mathbb{Z}}(\mathbb{Q}, C) = 0$ for every cyclic group $C$

This is part of an exercise I'm doing, exercise 2.22 Rotman, Introduction to homological algebra. Prove that $$\hom_{\mathbb{Z}}(\mathbb{Q}, C) = 0$$ for every cyclic group $C$. Any hint ?
0
votes
0answers
55 views

Describe the kernel and the fibers of $\phi$ geometrically (as subsets of the plane).

Define $\phi : \mathbb{C}^{\times} \mapsto \mathbb{R}^{\times}$ by $\phi(a+bi) = a^2 + b^2$. Prove that $\phi$ is a homomorphism and find the image of $\phi$. Describe the kernel and the fibers of ...
0
votes
1answer
23 views

Identifying some cyclic subgroup

Is there a fast way to argue that (for $a,b>1$ integers) the set of all $x\in\mathbf{Z}/b\mathbf{Z}$ with $ax=0$ is isomorphic to $\mathbf{Z}/{gcd(a,b)}\mathbf{Z}$? Maybe by counting the elements, ...
1
vote
1answer
64 views

Cohomology of finite p-groups

Given a finite abelian $p$-group $A$ acted on by a finite $p$-group $G$. Under the assumption $\operatorname{H}^1(G,A_1)=0$, where $A_1$ is the set of elements of $A$ having order at most $p$, what ...
2
votes
1answer
59 views

A module with 300 elements

I have got this problem. Let it be $R=M_{2}(Z)$ the ring of square matrices over the integers. I need to find a $R-$module with $300$ elements and one question for this problem, can be there a ...
4
votes
1answer
98 views

definition of (semi)group (co)homology

I'm puzzled why "group cohomology" contains terms 'group' (instead of 'semigroup') and 'cohomology' (instead oh 'homology and cohomology'). I'm new to the subject. Please inform me of any claims ...
7
votes
1answer
132 views

Computing the action of $S_3$ on $H^n(\mathbb{Z}_3,\mathbb{Z})$

Let $G=S_3$ and let $H$ be the Sylow $3$-subgroup in $G$. If $\mathbb{Z}$ is the trivial module, then it can be shown that $$H^n(H,\mathbb{Z})=\begin{cases}\mathbb{Z}&n=0\\0&n\text{ ...
3
votes
1answer
95 views

A short exact sequence of groups

I have a basic question from Rotman's 'An Introduction to Homological Algebra', Thm 5.3 pp 152. Let \begin{equation*} 0\xrightarrow{} A\xrightarrow{} E\xrightarrow{\pi} G\xrightarrow{} 1 ...
0
votes
0answers
43 views

A general form of $H_i(\Gamma,\mathbb{Z})$

Given a group $\Gamma (\subset PGL_d(K))$, is there some general form to express $H_i(\Gamma,\mathbb{Z})$ in terms of it?
8
votes
3answers
183 views

Is $G$ a semidirect product of $Z(G)$ and $\operatorname{Inn}(G)$?

The title pretty much sums it up. $\operatorname{Inn}(G)$ is the group of inner automorphisms, $Z(G)$ is the center. I know that $\operatorname{Inn}(G)$ is isomorphic to $G/Z(G)$. This means that ...
5
votes
2answers
76 views

A generalization of abelian categories including Grp

The category of groups shares various properties with abelian categories. For example, the Five lemma and Nine lemma hold in Grp. Is there a weakened notion of abelian category which also includes Grp ...
3
votes
1answer
96 views

Sufficient condition for a direct limit of abelian groups to be infinitely generated

I have the following setup. The CW-complexes $\Gamma_n$ are equipped with maps $\gamma_n\colon\Gamma_{n+1}\rightarrow\Gamma_{n}$ and it is known that the rank of their first cohomology groups is ...
4
votes
1answer
169 views

Groups acting on polytopes

I am currently reading the paper "Polytopal Resolutions for Finite Groups" [1] by Graham Ellis, James Harris and Emil Skoeldberg and have a question regarding an early remark of theirs. Their basic ...
7
votes
1answer
503 views

Equivalences and isomorphisms of short exact sequences

In case it's necessary, I'm working in the category $\mathbf{Ab}$ of abelian groups. My question concerns what I find to be a strange way of viewing the elements of the Ext group $\mbox{Ext}(A,B)$ of ...
0
votes
1answer
49 views

how to make factorization by a group action

any algebra and numerical example for Projectivization http://en.wikipedia.org/wiki/Projectivization which book or paper teaching this
1
vote
1answer
80 views

Projective group

I was reading about the torsion free abelian group. Does there exist a torsion free abelian group which is not projective but for which each of its torsion free homomorphic images are projective?
3
votes
1answer
130 views

Group extension: $H^n(\mathbb{Z},M)$

Let $M$ be any $\mathbb{Z}$-module. Find $H^n(\mathbb{Z}, M)$ for all $n$. Use different approaches for the case $n=2$. First approach: because $\mathbb{Z}$ is a free $\mathbb{Z}$-module, it is ...
1
vote
0answers
170 views

condition for short exact sequence of groups to be isomorphic

Let $G$ be a group and $K_1,K_2$ be two distinct normal subgroups of $G$. We have two short exact sequences: $$1 \to K_1 {\rightarrow} G {\rightarrow} G/K_1 \to 1$$ $$1 \to K_2 {\rightarrow} G ...
2
votes
1answer
65 views

Conditions for a $\mathrm{Hom}$ group to be finite.

If $G$ is a finite group, and $D$ is a divisible abelian group, what are some conditions on $D$ for which $\mathrm{Hom}(G,D)$ is finite? At first I thought that having $D$ with finite torsion ...
1
vote
2answers
184 views

cohomology of a finite cyclic group

I apologize if this is a duplicate. I don't know enough about group cohomology to know if this is just a special case of an earlier post with the same title. Let $G=\langle\sigma\rangle$ where ...
1
vote
1answer
381 views

Rank of a cohomology group, Betti numbers.

How is the rank of a cohomology group computed and what does it convey? I am trying to understand the concept behind betti numbers in a simplicial homology. Edited with details: Given a set of ...
3
votes
1answer
81 views

Map induced between Pontryagin duals

Let $f\colon A\to B$ be a group homomorphism between finite abelian groups. For abelian group $G$, let $G^\wedge=\operatorname{Hom}_\mathbb{Z}(G,\mathbb{Q}/\mathbb{Z})$ be its Pontryagin dual. Since ...
3
votes
1answer
268 views

Homology and semidirect products

If $G=N\rtimes H$ what is the relation between the second integral homology groups (Schur multipliers) of $G,N$ and $H$.
0
votes
1answer
120 views

A simple second homology question

What is $H_2(\mathbb{Q},\mathbb{Z})$ where the action is trivial. Thanks in advance
3
votes
1answer
357 views

how can we compute the homology of these groups without using topology?

I'd like to know the homology of a free group and a free abelian group of rank 2. I know that they could be computed topologically, but I'm searching a proof purely algebraic, could you help me ...
8
votes
6answers
731 views

Why are projective objects important?

I belive we study them because in important categories they are close to free objects and even a retract of a free object in some algebraic instances (for example, direct summands in Mod_R, and ...
13
votes
2answers
1k views

Short Exact Sequences & Rank Nullity

This is a well known lemma that consistently appears in textbooks, either as a statement without proof, or as an exercise (see for example pp. 146 of Hatcher) If $0 \stackrel{id}{\to} A ...