Normal rank two Sylow subgroup for least prime divisor has normal complement if the prime is odd

This article states and (possibly) proves a fact that is true for certain kinds of odd-order groups. The analogous statement is not in general true for groups of even order.
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This article describes a result about finite groups where the isomorphism types of the Sylow subgroup (?)s implies certain properties of the whole group. Note that all $p$-Sylow subgroups for a given prime $p$ are conjugate and hence are isomorphic, so the statement makes sense.
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This result relates to the least prime divisor of the order of a group. View more such results

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 $p$.
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Statement

Suppose $G$ is a finite group, and $p$ is the least prime divisor of the order of $G$. Suppose $P$ is a $p$-Sylow subgroup of $G$ and the normal rank of $P$ is at most two. Suppose further that $p$ is odd. Then, there exists a normal p-complement in $G$. In other words, there exists a normal subgroup $N$ of $G$ that is a permutable complement to $P$. Thus, $G$ is a P-nilpotent group (?).

Related facts

• Cyclic Sylow subgroup for least prime divisor has normal complement: This holds for odd primes as well as for the prime $2$.
• Cyclic normal Sylow subgroup for least prime divisor is central
• Normal of least prime order implies central

References

Textbook references

• Finite Groups by Daniel Gorenstein, ISBN 0821843427, Page 257, Theorem 6.1, Chapter 7 (Fusion, transfer and p-factor groups), section 7.6 (elementary applications), ^{More info}