A basic question on group cohomology

Question

I am studying group cohomology, and those constructs are a bit abstract to me, I can't even write down a concrete example. For example, I'm having some trouble understanding the following assertion: if $H$ is a normal subgroup of a group $G$, and $A$ is a $G$-module, then cohomology group $H^n(H,A)$ is a $G/H$-module. Let us consider the simplest case as follows: assume that $H$ acts trivially on $A$ and $n=1$, then $H^1(H,A)=\text{Hom}(H,A)$. So, how to define a $G/H$-action on $\text{Hom}(H,A)$ explicitly in this case?

Answer 1

The short answer to your question is that one defines $$(g\cdot\phi)(h)=\phi(g^{-1}hg)$$

But I'd like to say a couple of things about your comment on the abstractness of group cohomology. I've found two concrete example helped me considerably.

The first are the so-called wallpaper groups, the two-dimensional cystallographic groups. The module here is just $\mathbb{R}^2$, and the groups are discrete geometric symmetries. Howard Hiller wrote an article on this, "Crystallography and Cohomology of Groups", which won an MAA Writing Award. See here for more, including links.

Here's how that action looks in a specific case. Consider the lattice $\mathbb{Z}^2$ of all integral points in $\mathbb{R}^2$. Let $G$ be the group of symmetries, $T$ the normal subgroup of translations; $A=\mathbb{Z}^2$ is our module, and $T$ acts on $A$ via translation. Then if $g\in G$ is, say, a $90^\circ$ rotation of the lattice, $g^{-1}tg$ will be the translation rotated by $90^\circ$. Finally, if $\phi$ is homomorphism $T\to A$ that sends $t$ to $t(0,0)$, $g\cdot\phi$ is the homomorphism sending $t$ to $g(t(0,0))$, i.e., $(0,0)$ translated by $t$ and then rotated $90^\circ$ about the origin.

The second example is the carrying function of grade school arithmetic. See here for more on that.

Finally, a page by David Corfield offers ten intuitions about cohomology.