I came across this problem that says:
If $Z(G)$ denotes the centre of the group $G$, then the order of the quotient group $G/Z(G)$
can not be which of the following?
(a) $15,$
(b) $25,$
(c) $6,$
(d) $4.$
Could someone point me in the right direction (a certain theorem or property that I have to use)? Thanks everyone in advance for your time.
Best Answer
We have the following theorem (a consequence of Sylow's theorems):
Let $G$ be a group of order $pq$ where $p,q$ are primes such that $p < q$ and $p$ does not divide $q-1$. Then $G$ is cyclic.
If $G / Z(G)$ has order $15$ then $p=3$ and $q-1 = 5-1 = 4$ so that $p$ does not divide $q-1$. Hence choice (a) is not possible using Chris Eagle's comment:
If $G / Z(G)$ is cyclic it follows that $G$ is abelian (prove it) so that $Z(G) = G$ and hence $G/Z(G) = \{0\}$.
Hope this helps.