I'm returning from an exam on group-theory and there were 2 questions I couldn't solve (and still can't), so I'm asking here for any hint you could possibly give.

Let G be a group and H and K subgroups such that $|H| = n$, $|K| = m$ and $gcd(n, m) = 1$. Show that $H \cap K = \{e\}$.

I wish I could show you some of my attempts before hand, but they're all rubbish that didn't get me anywhere. Essentially, the only (and last) thing I remembered and thought it could be useful was to see if if H and K are partitions of G. I think I've read something similar somewhere but can't recall where, so, am uncertain about it.

The other question I couldn't solve is, I think, related to this, so I shall try it once I understand this one.

Thanks for taking the time to read! Any tip is appreciated.

  • 1
    $\begingroup$ What do you know about the relationship between the order of a group and the order of one of is subgroups? $\endgroup$ Commented Feb 3, 2011 at 17:06
  • $\begingroup$ The order of a group is a multiple of the order of its subgroups, by the Lagrange's theorem, right? So, if $|G| = 20$, $|H|$ and $|K|$ could be 4 and 5. But I fail to see a connection, yet. $\endgroup$
    – Marla
    Commented Feb 3, 2011 at 17:11

1 Answer 1


Let $g \in H\cap K$. By Lagrange's Theorem, the order of g divides m and n, but the greatest common divisor of m and n is 1, so g=e.

  • 8
    $\begingroup$ Please do not post complete answers to homework problems. $\endgroup$ Commented Feb 3, 2011 at 17:11
  • $\begingroup$ Great answer, and is $g=e$ because $|H\bigcap K|=1? $\endgroup$
    – grayQuant
    Commented Mar 2, 2015 at 2:33
  • $\begingroup$ @grayQuant The order of $g$ is 1, i.e. $g^1=g=e$. $\endgroup$ Commented Mar 2, 2015 at 8:52

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