Cyclic Sylow subgroup for least prime divisor has normal complement

This article gives the statement, and possibly proof, of a normal p-complement theorem: necessary and/or sufficient conditions for the existence of a Normal p-complement (?). In other words, it gives necessary and/or sufficient conditions for a given finite group to be a P-nilpotent group (?) for some prime number $p$.
View other normal p-complement theorems

This result relates to the least prime divisor of the order of a group. View more such results

History

This result is generally attributed to Burnside. It appeared in a paper by him published in 1895.

Statement

Suppose $G$ is a finite group and $p$ is the least prime divisor of the order of $G$. If a $p$-Sylow subgroup of $G$ is cyclic, then it has a normal complement: in particular, it is a retract. In symbols, if $P$ is a cyclic $p$-Sylow subgroup of $G$, then there exists a normal subgroup $N$ of $G$ such that $NP=G$ and $N\cap P$ is trivial. Another way of putting this is that $G$ is a P-nilpotent group (?).

Related facts

• Burnside's normal p-complement theorem
• Cyclic normal Sylow subgroup for least prime divisor is central
• Normal of least prime order implies central

Applications

• Order is twice an odd number implies subgroup of index two

Facts used

1. Sylow satisfies intermediate subgroup condition
2. Cyclic normal Sylow subgroup for least prime divisor is central
3. Burnside's normal p-complement theorem

Proof

Given: A finite group $G$. $p$ is the least prime divisor of the order of $G$, and $P$ is a $p$-Sylow subgroup of $G$.

To prove: $P$ has a normal complement in $G$.

Proof:

1. $P\leq Z(N_{G}(P))$: Consider $P$ as a subgroup of its normalizer $N_{G}(P)$. Note that $p$ is still the least prime divisor of the order of $N_{G}(P)$, and by fact (1), $P$ is $p$-Sylow in $N_{G}(P)$. Thus, by fact (2), $P\leq Z(N_{G}(P))$.
2. $P$ has a normal complement: This follows from the conclusion of the previous step and fact (3).

References

Expository references

• A brief history of the classification of the finite simple groups by Ronald Mark Solomon, Bulletin of the American Mathematical Society, ISSN 10889485 (electronic), ISSN 02730979 (print), Volume 38,Number 3, Page 315 - 352(Year 2001): An expository paper by Ronald Mark Solomon describing the 110-year history of the classification of finite simple groups.^{Weblink (PDF)More info}: Page 316 (relative page 2).

Journal references

• Notes on the theory of groups of finite order by William Burnside, Proceedings of the London Mathematical Society, ISSN 1460244X (online), ISSN 00246115 (print), Volume 26, Page 191 - 214(Year 1895): ^{More info}