let $D_8$ be the group of symmetries of a square. Not counting the trivial subgroup, how many distinct cyclic subgroup s does $D_8$ contain?
My result:
A group $D_8$, contains an element $a$ of order 4.
$<{a}> =(a^4:4\in Z)$
$(1.a,a^2,a^3,x,b,c,d)$ the reflections and symmetries in $D_8$.
$(1)(a)=a
\\
(a)(a)=a^2
\\
(a)(a^2)=a^3
\\
(a)(a^3)=a^4=1$
$(1,a,a^2,a^3)$
Did I answer this correctly? If yes can someone clarify my writing or if it's wrong please guide me …
Best Answer
You found one such cyclic subgroup.
We also have a cyclic subgroup that's also a subgroup of the cyclic group you found, of order two:$$\{1, a^2\}$$
Four additional cyclic subgroups of order two are as follows:
$$\{1, x\}, \{1, b\}, \{1, c\}, \{1, d\}$$
Of course, we need also to add a sixth, trivially cyclic but distinct subgroup: $\{1\}$.
So, with the subgroup you found, and the additional 6 subgroups here, we have, in all, $7$ distinct cyclic subgroups of $D_8$.