Looking for a little help in understanding 'orders of elements'

Asked Mar 18 '21 at 19:33

Active Mar 18 '21 at 21:51

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I'm currently working through a proof where I need to prove that a group $H$ is a subgroup of $G$ such that $\def\ord{\operatorname{ord}} H=\{g\in G \mid \ord(g) \text{ is a divisor of } 12\}$. I already know that $G$ is commutative, and that the general structure to prove a given group is a subgroup of another by proving that it's closed under multiplication, has association, has an identity element, and has an inverse. However, my main issue in proving this is a lack of understanding on the relationship that $\ord(g)$ has a divisor of $12$.

Any help in understanding the order of elements in context to this problem would be greatly appreciated!

Thank you in advance.

edited Mar 18 '21 at 21:51

Sammy Black

25,273

asked Mar 18 '21 at 19:33

4

Hint: another way to describe $H$ is that $H = {g \in G \mid g^{12}=e}$. – Théophile Mar 18 '21 at 19:42
What exactly do you have trouble understanding? – anon Mar 18 '21 at 22:39
I'm trying to prove that $H$ is a subgroup of $G$ using the rule above but I'm not sure how to write a proof with an order – Mar 19 '21 at 01:32
Hint: apply the subgroup test using $, g,h\in H\Rightarrow g^{12}=1=h^{12}\Rightarrow (g/h)^{12}=1\Rightarrow g/h\in H\ \ $ – Bill Dubuque Mar 19 '21 at 01:32
If you now understand it, please delete the question since it is a duplicate. – Bill Dubuque Mar 19 '21 at 01:33
Recall that $,a^{12}=1\iff ord(a)\mid 12\ \ $ – Bill Dubuque Mar 19 '21 at 01:37

Looking for a little help in understanding 'orders of elements'

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