## Analog to the Chinese Remainder Theorem in groups other than Z_n.

The Chinese Remainder theorem is usually thought of as an isomorphism of rings, not just of cyclic groups. In this …

## A morphism intertwining two induced representations

(1) is certainly false in general, but some things like it hold at least formally. See e.g. “Motivation” in Knapp …

## alternating and symmetric powers of the standard representation of the symmetric group

Let \$W = mathbb C^n\$, with the obvious representation of \$S_n\$. Lemma Let \$lambda\$ be a partition of \$n\$. The …

## A linearly orderable monoid which does not embed into a linearly orderable group

Malcev’s example is orderable. See https://doi.org/10.2307/2036896. So the answer is known and in the negative. There is a prominent example …

## Algorithm for root system of Coxeter group generated by permutations

There is a theoretical answer (as opposed to an algorithmic answer) found in Björner and Brenti’s “Combinatorics of Coxeter groups”, …

## \$A\$ is isomorphic to \$A oplus mathbb{Z}^2\$, but not to \$A oplus mathbb{Z}\$

Let \$A\$ be the additive group of bounded sequences of elements of \$mathbb{Z}[sqrt{2}]\$. Clearly \$Acong Aoplusmathbb{Z}[sqrt{2}]cong Aoplusmathbb{Z}^2\$ as abelian groups, …

## A generously vertex transitive graph which is not Cayley?

Take the graph product \$G = P times mathbb{Z}\$ of the Petersen graph with the infinite path graph. This is …

## A finite group \$G\$ all of whose reps are defined over \$mathbb{Z}\$ and yet \$Rep(G)\$ is not generated by permutation representations

The answer is no. Counterexamples include the Weyl groups of types E6, E7, and E8. For a proof that the …

## Acyclic group and finite CW-complex

I presume by “acyclic” you are referring to homology with \$mathbb{Z}\$ coefficients. There are many such examples. For instance, you …

## Abundancy index and non-solvable finite groups

I can answer Questions 1 and 4. Make sure you look at S. Carnahan’s answer. It deals with Questions 2 …