On relative normal complements in finite groups. III

Leonard, Henry S.

doi:10.1090/S0002-9939-1985-0796435-X

On relative normal complements in finite groups. III
HTML articles powered by AMS MathViewer

by Henry S. Leonard PDF

Proc. Amer. Math. Soc. 95 (1985), 5-6 Request permission

Abstract:

Let $G$ be a finite group, let $H \leq G$, and let $\pi$ be the set of prime divisors of $\left | H \right |$. Assume that whenever two elements of $H$ are $G$-conjugate then they are $H$-conjugate. Assume that for all $h \in {H^\# }$, $({C_G}(h):{C_H}(h))$ is a $\pi ’$-number. We prove that $H$ is a $\pi$-Hall subgroup and that there exists a normal complement ${G_0} = G - {({H^\# })^{G,\pi }}$. An example shows that the generalization to relative normal complements is not true.

References

Richard Brauer, On quotient groups of finite groups, Math. Z. 83 (1964), 72–84. MR 159872, DOI 10.1007/BF01111110
Henry S. Leonard Jr., On relative normal complements in finite groups, Arch. Math. (Basel) 40 (1983), no. 2, 97–108. MR 720899, DOI 10.1007/BF01192757
Henry S. Leonard Jr., On relative normal complements in finite groups, Arch. Math. (Basel) 40 (1983), no. 2, 97–108. MR 720899, DOI 10.1007/BF01192757
Geoffrey R. Robinson, Blocks, isometries, and sets of primes, Proc. London Math. Soc. (3) 51 (1985), no. 3, 432–448. MR 805716, DOI 10.1112/plms/s3-51.3.432