Suppose that G is a finite group and P is a Sylow p-subgroup of G, and H/K is a chief factor of G, and T/K is a Sylow p-subgroup of H/K. Is that correct to say that c(T/K) is less than or equal to c(P), where c(P) means nilpotency class of P ?
Dear Saeid,
suppose that |G| =m.p^a, where (m,p)=1, and |K|=k.p^b, where (k,p)=1, and K is a normal subgroup in G. Now consider a Sylow p-subgroup L in K, |L|=p^b. One can find (due to Sylow theorem) a Sylow p-subgroup PP in G, |PP|=p^a, such that L
I will try and reformulate in words the answer by Yuri Semenov.
You need to know first of all that if you go from a group to a subgroup, or to a quotient group, then the nilpotence class (NC) cannot increase.
You also need to know that a Sylow subgroup of a quotient group of the group $G$ is the image of a Sylow subgroup of $G$. And then, by one of the Sylow theorems, a Sylow subgroup of a subgroup of $G$ is a subgroup of a suitable Sylow subgroup of $G$.
With these premises, a Sylow subgroup of the subgroup $H$ of $G$ will have NC less than or equal to the NC of that of a Sylow subgroup of $G$. And then the same holds for a Sylow subgroup of the quotient group $H/K$.
Note that as I have written it, the argument applies also to a composition factor.
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