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I am studying an undergraduate course on group theory, and I have a question about the following remark made in the lecture notes:

The internal direct products are useful for helping us sort out the structure of a given group. External direct products are useful for constructing new groups from old groups.

I understand that the internal and external direct products are essentially the same thing (at least I think that's the case).

I can't find any examples where, given a group, simply our knowledge of the internal direct product helps us understand the structure of that group, unless we already know the subgroups which our group is the internal direct product of.

Does the quote above imply that we can use only our knowledge of the internal direct product to understand the structure of a group?

Or, does it actually mean that if the subgroups (which the group is the internal direct product of) are known, then we can understand the structure of the group?

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well, it helps for example when proving that every group of order $pq$ such that $p<q$ and $p\not\equiv 1 \bmod q$ is cyclic.

The way to do this is to notice that the $p$-sylow and $q$-sylow subgroups are normal by Sylows theorem and then noticing that if $P$ is the unique $p$-sylow subgroup and $Q$ is the unique $q$-sylow subgroup then $P\cap Q=\{e\}$ by lagrange's theorem.

It follows that $PQ$ is an internal direct product of order $pq$. Since $P$ and $Q$ must be cyclic and have coprime order it follows that $PQ$ is cyclic.

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  • $\begingroup$ Do I misunderstand your first statement? If I pick a group of order $pq$ where $p=2$ and $q=4$ then it is cyclic? $\endgroup$ – Matt Jan 2 '17 at 17:23
  • $\begingroup$ $p$ and $q$ are primes here. $\endgroup$ – Jorge Fernández Hidalgo Jan 2 '17 at 17:24
  • $\begingroup$ Got it. Thanks. $\endgroup$ – Matt Jan 2 '17 at 17:25

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