# Tagged Questions

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### 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 ...
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### how is the jordan-holder theorem used in conjunction with short exact sequences to construct groups of certain order?

I am an undergraduate and have been asked to explain how simple groups can be used to construct groups of finite order. I started with reading about the extension problem in group theory and from ...
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### What is explicit isomorphism between $H^2(G,\mathbb{Q}/\mathbb{Z})$ and $H_2(G,\mathbb{Z})$?

Let $G$ be a finite group. Then its Schur multiplier is the second cohomology group $H^2(G,\mathbb{Q}/\mathbb{Z})$, which is isomorphic to the second homology group $H_2(G,\mathbb{Z})$ (Proof can be ...
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### Understanding the Bockstein homomorphism in group cohomology

Let $k$ be an algebraically closed field of characteristic $p>0$. Let $G$ be a finite group. In group cohomology, the Bockstein homomorphism is the connecting homomorphism ...
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### Show that $H^{\prime} \cap A$ is a homomorphic image of $M(G)$

Let $H$ be a group and $A$ be a central subgroup of $H$ of finite index. Let $G =H/A$. Show that $H^{\prime} \cap A$ is a homomorphic image of $M(G)$. Here $H^{\prime}$ denotes the commutator ...
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### How do you compute group cohomology in practice?

If you have a finite group $G$ and a finite $G$-module $K$, and you need to know $H^1(G,K)$ or $H^2(G,K)$, how do you do it? Do you use a computer algebra system? (If so, which one?) Do you use a ...
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### Property of G-modules involving the invariant elements under the G-action

I am stuck at some basic fact I would like to prove. I tried proving it using $G-$orbits and cardinalities, but without success. Let $p$ be some prime number, $G$ be a finite $p-$group and $A$ a ...
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### Transgression homomorphism on cohomology

I have one confusion about the transgression homomorphism which I found in two different books. I am unable to show that they really are same. The first description of transgression homomorphism I ...
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### understanding into algebraic terms difference between homology and cohomology

my previous question understand quotient group was related to understanding of quotient group,i dont need to know too much detailed in group theore,just some part of algebraic topology,especially ...
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### Why is it sufficient to prove for $H^0$

I am reading the corestriction homomorphism on cohomology. To prove a proposition (for example the composition of corestriction and restrcition is the multiplication by the index), it says that it is ...
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### Show that $con_H^g$ is the identity map

Let $G$ be a group,$H$ a subgroup of $G$ and $A$ be $G$-module. Let $g \in G$ and $H^g=g^{-1}Hg$. Let $Z^2(G,A)$ be the set of $2$-cocycles. Given $f \in Z^2(H,A)$, let $f^g \in Z^2(H^g,A)$ be defined ...
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### Problem with cohomology (II)

Let $G$ be a group and $K$ be subgroup of $G$, Let $A$ be $G$ module. Let $X=\{X_n\}$ be free resolution of $\mathbb{Z}$. Let $\delta$ represents collectively the homomorphism induced in Hom sequence ...
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