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The first example that comes to mind, $G=\bigoplus_{i=1}^\infty\mathbb{Q}$ and $H=\mathbb{Q}/\mathbb{Z}\oplus\bigoplus_{i=1}^\infty\mathbb{Q}$, seems to work. FYI, a related example $G=\bigoplus_{i= …

This question probably has a simple and immediate answer which escapes me now. (And, I should admit, it's more my curiosity than anything else.) The only natural way to construct a group structure on …

I am a bit puzzled by your question. Do you mean the Lagrange's theorem stating that the order of a subgroup divides the order of the group? In that case, even for the semigroups defined in your secon …

This question on subgroups of $GL(2,q)$ asked by Jan, and especially wonderful answers to it given by Geoff Robinson, Ralph, and Will Sawin showing that "almost no finite groups" inject in $GL(2,q)$ m …

This is follow-up to my earlier question. Suppose that we have elements $\sigma_1,\ldots,\sigma_k\in S_n$, and that we established that these elements actually generate $S_n$. Since that previous qu …

Somehow this question made me think of instances of small exceptions in general, and I remembered the statement I heard once that $S_5,A_6,S_6,A_7,A_8,S_8$ are the only instances of symmetric/alternat …

I just realised that the meaning of the term "metabelian", when applied to groups, or Lie algebras, seems to have changed over years. (These days, it means that $[[G,G],[G,G]]$ is trivial, while in th …

My favourite example is related to some bit of work of my own, so I apologise in advance for a bit of self-promotion. It concerns dealing with symmetric operads (algebraic ones, meaning that the $n$-t …

For the reasons apparent below I shall use the notation $C_N(m,j)$, not $C(m,j)$. It is sufficient to prove $$ \sum_{m=0}^N\sum_{j=0}^m jC_N(m,j)x^m=Nx\cdot x(x+1)\cdots (x+N-2), $$ as this formula …

One reference where the asymptotic result you are asking for was first established (I think), as well as some reasonable growth estimates for $a_n$, is D.G.Kendall and R.A.Rankin, "On the number of A …

I am not quite sure about the reference :( I always thought of this fact as follows. Matrix elements of tensor powers of a representation U are all possible monomials in matrix elements of U, so the …

This question sounds like it should be very well known, but for some reason I failed to find a decent answer anywhere. Let $G\subset\mathbb{Z}^n$ be a subgroup, and $G_+=G\cap\mathbb{Z}_{\ge0}^n$ be a …

In GAP (https://www.gap-system.org), there is a function IsSymmetricGroup, which tells you whether a subgroup of $S_n$ generated by given permutations is all of the $S_n$. It looks like it takes virtu …

The fact that the Lie module (as proposed by Darij Grinberg) works, as well, as an explicit isomorphism of modules, follows from the theory of cyclic operads: see Corollary 6.9 in http://sites.math.no …

Euclidean domain: Z[i] (Gaussian integers) Principal ideal domain: ring of integers in Q(\sqrt{-19}) Unique factorization domains: Z[x], C[x,y] Finite field: F_4=F_2[t]/(t^2+t+1)