Cyclic Sylow subgroup for least prime divisor has normal complement
From Groupprops
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 .
View other normal p-complement theorems
This result relates to the least prime divisor of the order of a group. View more such results
Contents
History
This result is generally attributed to Burnside. It appeared in a paper by him published in 1895.
Statement
Suppose is a finite group and is the least prime divisor of the order of . If a -Sylow subgroup of is cyclic, then it has a normal complement: in particular, it is a retract. In symbols, if is a cyclic -Sylow subgroup of , then there exists a normal subgroup of such that and is trivial. Another way of putting this is that 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
Facts used
- Sylow satisfies intermediate subgroup condition
- Cyclic normal Sylow subgroup for least prime divisor is central
- Burnside's normal p-complement theorem
Proof
Given: A finite group . is the least prime divisor of the order of , and is a -Sylow subgroup of .
To prove: has a normal complement in .
Proof:
- : Consider as a subgroup of its normalizer . Note that is still the least prime divisor of the order of , and by fact (1), is -Sylow in . Thus, by fact (2), .
- 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}