New posts in Finite Groups

Let $G$ be a group with $|G|=102 ( = 2 \cdot 3 \cdot 17)$, with $|Z(G)| = 2$. Prove the following:

Is every direct product of (finite) cyclic groups abelian?

Does $G$ being a subgroup of $GL(n, \mathbb Z/p\mathbb Z)$ for all odd primes $p$ imply it is a subgroup of $GL(n, \mathbb{Z})$?

The symmetry of conjugacy classes of a subgroup of a finite group

Give a reason why $\mathbb Z_{64}\times \mathbb Z_4$ and $\mathbb Z_{64}\times \mathbb Z_{2}\times \mathbb Z_{2}$ are not isomorphic.

Elements of the same order in the same conjugacy class that commute are contained in the same cyclic subgroup

Exercise 6.17 from Isaac's Character Theory of Finite Groups.

Are there any finite examples of metachirality?

Is a group isomorphic to the internal product of its Sylow p-subgroups?

Let $G$ be a finite abelian group, and let $n$ divide $|G|$. Let $m$ be the number of solutions of $x^n=1$. Prove that $n\mid m$.