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It is the set-wise stabilizer of $S$ under the left regular action.

I'm a little hesitant to say anything in the face of all the comments above, but I think Ana is asking for an answer to the following question: If $G$ is a reductive algebraic $k$-group, and we choose …

It appears to be a $p$-adic analog of the lamplighter group.

Computational evidence suggests $g(n)$ varies wildly with $n$. When $n$ is a power of a prime, or has lots of small factors, $g(n)$ can be very large (I would guess $g(2^k)$ is superexponential in $k …

I think the claim goes back to the 1979 paper "Monstrous Moonshine" by Conway and Norton, where they discuss the "defining property of 24": If $n$ is a positive integer such that $xy \equiv 1$ mod $n$ …

Any subgroup of $H \times K$ will have a subgroup of index at most 4 that is a product of two cyclic groups. This comes from intersecting with a fixed choice of cyclic subgroups of index 2 in each fa …

A permutation group is called regular if it is transitive and all stabilizers are trivial. Left multiplication yields a regular embedding of any group into its group of permutations (so the answer to …

The commutator subgroup of a group is given by a functor on the category whose objects are groups and whose morphisms are all homomorphisms. We can say the similar statement for the torsion subgroup …

I might as well add another two interpretations of this group from the theory of algebraic tori - this should make the isomorphism between $SO_2(\mathbb{Q})$ and $\mathbb{Q}[i]^\times/\mathbb{Q}^\time …

The holomorph is used in the theory of Hopf Galois structures. If L/K is a Galois extension with Galois group G, then isomorphism classes of K-Hopf algebras H such that there is an L-Hopf algebra iso …

From Christopher Perez's answer, we have $|Sp_{2n}(\mathbb{Z}/p\mathbb{Z})| = p^{n^2} \prod_{i=1}^n (p^{2i}-1)$. Following Johannes Hahn, we wish to determine the size of the kernel of the homomorphi …

I can't give you a complete answer (perhaps an expert will come along...), but a Google search reveals that there is an exercise on page 213 of Mac Lane and Moerdijk, Sheaves in geometry and logic: a …

The only theorem of Tomkinson I could find about this was in Tomkinson, M.J.. "Groups as the union of proper subgroups." Mathematica Scandinavica 81.2 (1998): 191-198. In section 3, he proves th …

I suggest you give Mariano a check mark. MathOverflow does not function properly when you change a question substantially after it has been correctly answered. Regarding your revised question, we ca …

If $m$ is big, you get a large family of nonabelian unipotent algebraic groups over $\mathbb{F}\_p$, and these yield the nonabelian p-groups. The standard examples include the group of $k \times k$ u …