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A subgroup $H$ of index $3$ determines a homomorphism $\text{SL}_2(\mathbb{F}_p) \to S_3$ whose kernel $N = \bigcap_{g \in \text{SL}_2(\mathbb{F}_p)} gHg^{-1}$ is a normal subgroup of index either $3$ …

This follows from the fact that the image of $\Gamma(2)$ in $\text{PSL}_2(\mathbb{Z})$ is freely generated by the two matrices you describe. There is a geometric proof of this fact based on the fact …

Here is an idea. Fix a commutative ring $R$ and elements $q, z \in R$. Recall that the (Iwahori-)Hecke algebra $H_n(q, z)$ is the $R$-algebra on generators $T_1, ... T_{n-1}$ with relations $$T_i T …

Yes; this is known as the longest element, and it exists and is unique for every finite Coxeter group (including the ones which do not arise as Weyl groups). The length of the longest element is the …

Okay, so let's fill in the details on Ville's nice argument in the comments: there is no such bound, and to prove this it suffices to exhibit a sequence of finite perfect groups whose ranks are unboun …

See Humphreys, starting from II.5.3. Given a Coxeter system $(W, S)$ let $V$ be the free vector space on symbols $\alpha_s, s \in S$. We define a bilinear form on $V$ by extending $$B(\alpha_s, \alph …

If $H$ is abelian, any homomorphism $G \to H$ factors through the abelianization $G/[G, G] \to H$, so we may assume WLOG that $G$ is also abelian, so we can apply the structure theorem to both $G$ and …

Here's a comment which might as well be written down. If $f$ is required to be an inner automorphism, then for $G$ finite this question can be understood using the character table of $G$: $x$ is c …

Any automorphism of $\mathbb{Z}_p$ preserves whether an element is divisible by $p^k$, so it is Lipschitz (in particular, continuous) with respect to the $p$-adic norm. On the other hand, any automor …

This is an elaboration on Stanley's reference to Exercise 5.13 in EC2. In these two blog posts you can find proofs using groupoid cardinality of the following results. For a finitely generated group $ …

Let $G$ be a group (if it helps, assume that $G$ is a Lie group or finite). Is a pair of elements $(g, h) \in G \times G$ determined up to simultaneous conjugacy by the conjugacy class of every elemen …

Yes. Some Googling turns up J. M. Tyrer Jones, "Direct products and the Hopf property," J. Austral. Math. Soc. 17 (1974), 174-196.

I can at least propose a definition. To my mind, the categorically best behaved category of graphs is the category of presheaves on $\{ \bullet \rightrightarrows \bullet \}$ ("directed multigraphs"); …

This is true iff $G$ is finite and abelian, the characteristic of $K$ does not divide $G$, and $K$ has all $n^{th}$ roots of unity whenever $G$ has an element of order $n$. Hopefully it is clear why $ …

The fusion ring, as a ring with basis, contains the same information as the character table. So your question, phrased in language more familiar to finite group theorists, is: Is a finite simple g …