Suppose |G| = 16 and Z(G) contains an element of order 4. What is the isomorphism type of G/Z(G)? I am sure that you have to use the fundamental theorem of abelian groups and the G/Z(G) theorem but I am not sure how to apply these to the problem?
Suppose |G| = 16 and Z(G) contains an element of order 4. What is the isomorphism type of G/Z(G)? I am sure that you have to use the fundamental theorem of abelian groups and the G/Z(G) theorem but I am not sure how to apply these to the problem?
SO $\displaystyle |Z(G)|\geq 4\Longrightarrow \left|G/Z(G)\right|\leq \frac{16}{4}=4$.
Now, it can't be $\displaystyle |Z(G)|=8$ since then $\displaystyle \left|G/Z(G)\right|=2\Longrightarrow G/Z(G)$ is cyclic non-trivial, which is impossible (this itself is a nice problem!), so it must be
either $\displaystyle |Z(G)|=16\iff G/Z(G)=1\iff G$ is abelian, or else $\displaystyle |Z(G)|=4\Longrightarrow G/Z(G)\cong \mathbb{Z}_2\times \mathbb{Z}_2$ since, as mentioned above, this factor
group cannot be cyclic non-trivial.
Tonio