Tessellate S.T.E.M.S - Mathematics - College - Set 2 - Problem 2

Algebra Level 3

For a finite group $G$ , we define $\alpha (G)$ to be the smallest natural number $n$ such that $G$ is isomorphic to a subgroup of $S_n$ , the permutation group on $n$ symbols. Which of the following options is correct for group $G$ with $n$ elements?

$(a)$ $\alpha (G) \leq n/2$ for all but finitely many natural numbers $n$ .

$(b)$ $\alpha (G) = n$ for infinitely many natural numbers $n$ .

$(c)$ $\alpha (G) \leq n/2$ for only finitely many natural numbers $n$ .

$(d)$ None of the above

This problem is a part of Tessellate S.T.E.M.S.

c b d a

1 solution

Patrick Corn
Jan 2, 2018

By Lagrange's theorem , $|G|$ must divide $\alpha(G)!.$ If $|G|=p$ is prime, then this implies that $\alpha(G)$ must be at least $p$ ; and if $|G|=p$ then $G$ is cyclic and hence isomorphic to a subgroup of $S_p$ generated by a $p$ -cycle. So if $|G|=p$ is prime, then $\alpha(G)=p.$ There are infinitely many primes, so the answer is (b).

Tessellate S.T.E.M.S - Mathematics - College - Set 2 - Problem 2

1 solution

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