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### Is there a formula for the number of elements in $S_n$ having length $k$ with respect to the generators taken to be the transpositions?

Define the length of a permutation as the minimum number of adjacent transpositions needed to describe it. We know that there is only one element of length $0$ in $S_n$ and $n-1$ elements of length $1$...
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### Transitive permutation groups which all of their proper subgroups are intransitive

Let $G$ be a transitive permutation group on a finite set $\Omega$. It is clear that if $G$ is regular, then every proper subgroup of $G$ is intransitive. Is there any other class of groups with this ...
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### Classification of indecomposable inclusions $(H \subset G)$ with $G$ decomposable

Definition: A group $G$ is indecomposable if: $G = G_1 \times G_2 \Rightarrow \exists i \ G_i = 1$. We can generalize the notion of indecomposable from groups to inclusion of groups as ...
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### Uniqueness of the direct product decomposition of inclusions of finite groups

This post is a generalization of Uniqueness of the direct product decomposition of finite groups. Here we look inclusions of finite groups $(H \subset G)$ instead of just finite groups. Definition: ...
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### What's the ratio of inclusions of finite groups with a distributive lattice?

Definition: Two inclusions of finite groups are equivalent, $(A \subset B) \sim (C \subset D)$, if: $(A/A_B \subset B/A_B) \simeq (C/C_D \subset D/C_D)$ with $A_B$ the normal core of $A$ in $B$. ...
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### Generalization of Frobenius groups

Frobenius group is a transitive permutation group on a finite set, such that no non-trivial element fixes more than one point and some non-trivial element fixes a point. In other words, if in a ...
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### When is the product of two permutations cyclic?

Call a subset $H \subset S_n$ "simple" if for every $q \in S_n$, there is some $p \in H$ such that $pq$ is cyclic (i.e. consists of a single cycle of length $n$). Is there some characterization known ...
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### Does this class of groups contain finitely generated infinite periodic groups?

Let $r(m)$ denote the residue class $r+m\mathbb{Z}$, where $0 \leq r < m$. Given disjoint residue classes $r_1(m_1)$ and $r_2(m_2)$, let the class transposition $\tau_{r_1(m_1),r_2(m_2)}$ be the ...
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### When is $S_n \times S_m$ a subgroup of $S_p$?

I asked the following question on math.stackexchange several months ago: Let $n,m,p>1$ be such that $S_n \times S_m \hookrightarrow S_p$. Does it imply that $p \geq n+m$? Derek Holt gave a ...
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### Question about doubly transitive groups with an n-cycle

Let $G$ be a doubly transitive subgroup of $S_n$ which contains an $n$-cycle, and let $G_{12}$ be the subgroup of $G$ consisting of all elements $g\in G$ for which $g(1)=1$ and $g(2)=2$. Then $G_{12}$...
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### Orbits of stabilizer of two points in a 2-transitive permutation group

I was doing something which needs to know sizes of all orbits of the stabilizer of two points in a 2-transitive permutation group. Since all 2-transitive permutation groups are known ans so are their ...
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### Primitive, non-2-transitive groups with very large orbitals?

Let $G$ be a transitive permutation group on a set $X$ with $n$ elements. Assume that $G$ is primitive, i.e., $G$ preserves no non-trivial partition of $X$. Assume as well that $G$ is not $2$-...
Let $G$ be a group and $H$ a subgroup such that there is a unique (non-trivial) intermediate subgroup $K$ (i.e. $H < S < G$ implies $S=K$) and $(H \subset K) \sim (K \subset G)$ (homogeneity) ...