0
$\begingroup$

I found the following question in an old test paper:

Suppose $H$ and $K$ are two subgroups of $G$ such that $|H|=3$ and $|K|=5$. Prove that $H\cap K = \{e_G\}$ where $e_G$ is the identity of $G$.

My attempt:

Every group of prime number order is isomorphic to $\Bbb Z_{p}$ ($p$ being the prime number) and thus they are necessarily cyclic. Thus, $H$ is isomorphic to $\Bbb Z_3$ and $G$ is isomorphic to $\Bbb Z_5$. After this I'm not sure how to proceed. I thought $\Bbb{Z_3}$ and $\Bbb{Z_5}$ have no element in common since the elements of $\Bbb{Z_3}$ are $\{\bar{0}_3,\bar{1}_3,\bar{2}_3\}$ while the elements of $\Bbb{Z_5}$ are $\{\bar{0}_5,\bar{1}_5,\bar{2}_5,\bar{3}_5,\bar{4}_5\}$. But again, I'm confused because since $H$ and $K$ are subgroups they must be sharing the identity element of $G$.

Questions:

So, could someone please clarify this problem of mine? Clearly $\Bbb Z_3$ and $\Bbb Z_5$ have no element in common, but then how do $H$ and $K$ have an element (the identity in common)?

Moreoever, what would the correct way to approach the quoted problem?

$\endgroup$
  • $\begingroup$ Please note that one the main parts of the question is: $H$ is isomorphic to $\Bbb Z_{3}$ and $K$ is isomorphic to $\Bbb Z_{5}$. But then $\Bbb{Z_3}$ and $\Bbb {Z_5}$ do not share their identity element. So how can $H\cap K$ have the same identity (given that the groups they are isomorphic to do not share an identity)? $\endgroup$ – user563280 May 21 '18 at 16:24
  • 1
    $\begingroup$ $H$ and $K$ occur in this question as subgroups of a group $G$, so they have the same identity. $\endgroup$ – Edward Evans May 21 '18 at 16:25
1
$\begingroup$

Any element of $H\cap K$ distinct from $e_G$ will have both order $3$ and $5$, but no such element of $g$ exists. To be more precise, if $g\in G$ is such that $g^3=g^5=e_G$, then $g=e_G$.

| cite | improve this answer | |
$\endgroup$
  • $\begingroup$ Thanks, but you missed a part of my question. $H$ is isomorphic to $\Bbb Z_{3}$ and $K$ is isomorphic to $\Bbb Z_{5}$. But then $\Bbb{Z_3}$ and $\Bbb {Z_5}$ do not share their identity element. So how can $H\cap K$ have the same identity (given that the groups they are isomorphic to do not share an identity)? $\endgroup$ – user563280 May 21 '18 at 16:22
  • $\begingroup$ @Blue The groups $\mathbb{Z}_3$ and $\mathbb{Z}_5$ don't heve a common element indeed, but how do you deduce from that that $H$ and $K$ don't have a common element? $\endgroup$ – José Carlos Santos May 21 '18 at 16:25
  • $\begingroup$ @Blue $H$ and $K$ are subgroups of the same group, so they do share a common identity, even if they are isomorphic to other groups that do not. $\endgroup$ – Bungo May 21 '18 at 16:27
  • $\begingroup$ @Bungo I see. I was making a wrong assumption. Thanks $\endgroup$ – user563280 May 21 '18 at 16:30
6
$\begingroup$

$H\cap{}K$ is a subgroup of both $H$ and $K$ and thus by Lagrange’s theorem its order divides both 3 and 5 thus it has to be just the identity element. The identity element in both groups is the same as in G because they are subgroups and the identity element of a group is unique.

| cite | improve this answer | |
$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy