Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Suppose G has a cyclic normal subgroup $\langle a\rangle$ of order $m$ and prime power index $s$ such that $m$ and $s$ are relatively prime. Then the following exact sequence splits:

$$1 \longrightarrow \langle a\rangle \longrightarrow G \longrightarrow G/\langle a\rangle \longrightarrow 1$$

Such group G is called a hyperelementary group.

Question: How to define a homomorphism $G/\langle a\rangle \rightarrow G$ to make the above sequence split ?

share|cite|improve this question
Use Sylow's theorem. – Jack Schmidt Jul 7 '11 at 14:14
Yes. And the splitting works for all normal subgroups of $p$-power index which have order prime to $p$, whether they are cyclic or not. Rather deeper is the Schur-Zassenhaus theorem, which says that a normal subgroup whose order and index are relatively prime is complemented. – Geoff Robinson Jul 7 '11 at 14:37
oh, I see, Thanks! – Yubin Jul 7 '11 at 14:59

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.