Abstract Algebra: Theory and Applications · Problem 13.27

Group Actions — Thomas Judson

Mathematics — Abstract Algebra · solvable-groups

Problem

Zassenhaus Lemma. Let

H

and

K

be subgroups of a group

G

. Suppose also that

H^*

and

K^*

are normal subgroups of

H

and

K

respectively. Then (a)

H^*(H \cap K^*)

is a normal subgroup of

H^*(H \cap K)

. (b)

K^*(H^* \cap K)

is a normal subgroup of

K^*(H \cap K)

. (c)

H^*(H \cap K)/H^*(H \cap K^*) \cong K^*(H \cap K)/K^*(H^* \cap K) \cong (H \cap K)/(H^* \cap K)(H \cap K^*)