A question of Malinowska on sizes of finite nonabelian simple groups in relation to involution sizes

Let $I_n(G)$ denote the number of elements of order $n$ in a finite group $G$. Malinowska recently asked “what is the smallest positive integer $k$ such that whenever there exist two nonabelian finite simple groups $S$ and $G$ with prime divisors $p_1,\,\cdots ,\,p_k$ of $|G|$ and $|S|$ satisfying $2=p_1<\,\cdots \

A question of Malinowska on sizes of finite nonabelian simple groups in relation to involution sizes

Let $I_n(G)$ denote the number of elements of order $n$ in a finite group $G$. Malinowska recently asked “what is the smallest positive integer $k$ such that whenever there exist two nonabelian finite simple groups $S$ and $G$ with prime divisors $p_1,\,\cdots ,\,p_k$ of $|G|$ and $|S|$ satisfying $2=p_1<\,\cdots \

A question of Mazurov on groups of exponent dividing 12

Mazurov asked whether a group of exponent dividing 12, which is generated by x, y and z subject to the relations $$x^3=y^2=z^2=(xy)^3=(yz)^3=1$, has order at most 12. We show that if such a group is finite, then the answer is yes

Influence of the number of Sylow subgroups on solvability of finite groups

Let $G$ be a finite group. We prove that if the number of Sylow $3$-subgroups of $G$ is at most $7$ and the number of Sylow $5$-subgroups of $G$ is at most $1455$, then $G$ is solvable. This is a strong form of a recent conjecture of Robati

Influence of the number of Sylow subgroups on solvability of finite groups

Let $G$ be a finite group. We prove that if the number of Sylow $3$-subgroups of $G$ is at most $7$ and the number of Sylow $5$-subgroups of $G$ is at most $1455$, then $G$ is solvable. This is a strong form of a recent conjecture of Robati