In other words we have to prove that and are well-defined, where denotes the largest normal pi-subgroup of G and denotes the smallest normal subgroup H of G such that G/H is a pi-group.
Thank you very much for your help in advance!
Let G be a finite group, a set of primes, the set of normal -subgroups of G, and the set of normal subgroups X of G with G/X a -group. Prove
(1) If H, K then HK . Hence is the unique maximal member of .
(2) If H, K then . Hence is the unique minimal member of
I would really appreciate your help.
In other words we have to prove that and are well-defined, where denotes the largest normal pi-subgroup of G and denotes the smallest normal subgroup H of G such that G/H is a pi-group.
Thank you very much for your help in advance!