Is the Tensor/Exterior square G⊗GG\otimes G or G∧GG\wedge G of infinite p-group also a p-group?

Let G be an infinite countable p-group. Is it true that G⊗G or G∧G are also p-groups? (where G acts on itself by conjugation). For simplicity, you can consider that G=[G,G], and then the above groups are isomorphic to the Universal Schur Cover. I am very frustrated, because a lot of websites and papers say … Read more

Computing the order of elements in a non abelian exterior square of a finite group

If we have an explicit group G, and we pick two elements g,h∈G, could we find the order of the element g∧h∈G∧G? The best thing I could find is Theorem 1.1 in Ellis’ Book (, which tells us that for any finite group G, H2(G) is a finite abelian group with exponent e dividing the … Read more

When is the restriction map $res:H^2(G,U(1))\to H^2(Z_p\times Z_p,U(1))$ not the zero map?

Consider $G$ to be a finite group with non-trivial Schur Multipler $H^2(G,U(1))$, where $G$ acts trivially on the circle group $U(1)$. By Example of a Schur-nontrivial group with no abelian subgroup of the form $H\times H$?, $G$ must have a subgroup of the form $Z_p\times Z_p$ for some prime $p$. My question is: under what … Read more

Pairing in Group Cohomology [closed]

Closed. This question is off-topic. It is not currently accepting answers. Want to improve this question? Update the question so it’s on-topic for MathOverflow. Closed 5 years ago. Improve this question I am following Ararat Babakhanian’s Cohomological Methods in Group theory. Let A,B,C be G modules then we have a G module structre on HomZ(B,C) … Read more

Generators of the symmetric square of the group ring of an abelian group

Let A be an abelian group and R=Z[A]– its group ring. Denote by I an ideal of R given by a kernel of the map R⟶Z⊕A, sending [r] to (1,r). Next, denote by S2I the symmetric square of I as an abelian group, tensored with Q. I claim that the elements ([a]+[b]−[c]−[d])2 with ab=cd generate … Read more

The fiber of the alternating map $X^{2n}\to \mathbb{Z}[X]$

Let $X$ be a fibrant connected simplicial set. There is a simplicial map $h_n\colon X^{2n}\to \mathbb{Z}[X]$, defined on points by $(x_1, \ldots x_{2n})\mapsto \sum\limits_{i=1}^{2n}(-1)^ix_i$. Here $\mathbb{Z}[X]$ is the simplicial abelian group associated to $X$ (the object that appears in the statement of Dold-Thom theorem). There is a natural map $f\colon F_{h_n} \to h_n^{-1}(0)$ between the … Read more

Homology of Torelli subgroup groups of automorphism groups of free products

For a group G, let G∗n denote the n-fold free product. There is a natural map Aut((Z/kZ)∗n)↦GLn(Z/kZ). Is it known if the group homology of the kernel of this map is finitely generated in each homological degree? This is false if you replace Z/kZ with Z. Answer AttributionSource : Link , Question Author : qqqqqqw … Read more

Quasi-inverse and homotopy invariance of Shapiro’s lemma map

I and two of my colleagues are currently wrestling with Shapiro’s lemma in the following situation. Let G be a finite group, let H be a subgroup of G, let k be an algebraically closed field of characteristic zero, and k∗ its multiplicative group. We consider the standard normal complexes C∙(G,IndGH(k∗)) and C∙(H,k∗), where IndGH(k∗)={a:G→k∗|a(xh)=h−1.(a(x)) ∀x∈G ∀h∈H}, … Read more

from 2-cocycle to classifying map

Let A,E,G:Set∗→Grp∗ be functors from pointed sets to (discrete) groups (∗=1) together with natural transformations i:A→E, p:E→G such that for any set X A(X)i→E(X)p→G(X) be a central extension which correspond to a specific (functorial) class [c]∈H2(G;A). Now we can extend these functors to sSet∗, then our central extension will become a principal fibration, which determined by … Read more