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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

Contents

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

Facts used

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 \le 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 \le 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

Journal references