A question about permutation and cyclic group

Question

Let $S_4$ be the group of permutations on $\{1,2,3,4\}$ and let $G = S_4\oplus \Bbb Z_4$. Find the order of the largest cyclic subgroup of $G$.

Answer 1

A subgroup of $G=S_4\oplus \Bbb Z_4$ is cyclic if it is generated by just one element. Let $G, H$ be finite groups, and let $g \in G, h \in H$. Then the order of the element $(g,h) \in G \oplus H$ is equal to the least common multiple of the orders of $g$ in $G$ and $h$ in $H$. The elements of $S_4$ have orders $1, 2,3,4$. The elements of $\Bbb Z_4$ have orders $1,2,4$. So we seek to maximize $lcm(a,b)$, where $a \in \{1,2,3,4\}, b \in \{1,2,4\}$. It is easy to see that the maximum value we may obtain is $12=lcm(3,4)$, so by choosing an element $g$ of order $3$ in $S_{4}$, $g=(123)$ for example, and an element $h$ of order $4$ in $\Bbb Z_4$ (e.g. $1 \in \Bbb Z_{4}$ assuming additive notation), the order of $(g,h)=((123),1) \in S_4\oplus \Bbb Z_4$ is $12$. So the order of the largest cyclic subgroup is $12$. $\square$