Abstract
John Tate [1] proved that if P â S y l p ( G ) P \in {\mathbf {S}}{\text {y}}{{\text {l}}_p}(G) , H H is a normal subgroup of a finite group G G and P ⊠H ⤠Ό ( P ) P \cap H \leq \Phi (P) ( ÎŚ ( G ) \Phi (G) is the Frattini subgroup of G G ) then H H has a normal p p -complement. We prove in this note that Tateâs theorem has nice character-theoretic applications.
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