New posts in Finite Groups

Extending the action $S_5$ on $2$-subsets of $\{1,\cdots,5\}$ to an action of $S_6$.

Let $G$ be a group of order $24$ having no normal subgroups of order $3$. Show that $G$ has four subgroups of order $6$.

Does a nontrivial finite solvable group have a subgroup of prime power index for each prime divisor?

Does the center of a perfect group not contain all elements of prime order?

Abelian finite groups and their subgroups

Let $G$ be a group with $33$ elements acting on a set with $38$ elements. Prove that the stabilizer of some element $x$ in $X$ is all of $G$.

(Proof verification) For each $s \in S$, show that $\sum_{t \in Gs} \frac{1}{|Gt|}=1$

Index of subgroups in a finite solvable group, with trivial Frattini subgroup (Exercise 3B.12 from Finite Group Theory, by M. Isaacs)

Are computer integers a finite group (under addition with overflow)?

Rotman's Introduction to the Theory of Groups Exercise 1.27