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 -Sylow subgroups for a given prime 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 .
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Contents
Statement
Suppose is a finite group, and is the least prime divisor of the order of . Suppose is a -Sylow subgroup of and the normal rank of is at most two. Suppose further that is odd. Then, there exists a normal p-complement in . In other words, there exists a normal subgroup of that is a permutable complement to . Thus, 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 .
- 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}