cmi-entrance 2017 Q4

cmi-entrance · India · pgmath 4 marks Not Maths
For a positive integer $n$, let $S_n$ denote the permutation group on $n$ symbols. Choose the correct statement(s) from below:
(A) For every positive integer $n$ and for every $m$ with $1 \leq m \leq n$, $S_n$ has a cyclic subgroup of order $m$;
(B) For every positive integer $n$ and for every $m$ with $n < m < n!$, $S_n$ has a cyclic subgroup of order $m$;
(C) There exist positive integers $n$ and $m$ with $n < m < n!$ such that $S_n$ has a cyclic subgroup of order $m$;
(D) For every positive integer $n$ and for every group $G$ of order $n$, $G$ is isomorphic to a subgroup of $S_n$.
For a positive integer $n$, let $S_n$ denote the permutation group on $n$ symbols. Choose the correct statement(s) from below:\\
(A) For every positive integer $n$ and for every $m$ with $1 \leq m \leq n$, $S_n$ has a cyclic subgroup of order $m$;\\
(B) For every positive integer $n$ and for every $m$ with $n < m < n!$, $S_n$ has a cyclic subgroup of order $m$;\\
(C) There exist positive integers $n$ and $m$ with $n < m < n!$ such that $S_n$ has a cyclic subgroup of order $m$;\\
(D) For every positive integer $n$ and for every group $G$ of order $n$, $G$ is isomorphic to a subgroup of $S_n$.