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Let G be a finite group and N a normal subgroup of G. Assume further that N is a p -group for some prime p.

1) By considering G/N, show that there is a subgroup H of G contaning N such that p does not divide [G:H].

2) Show that N is a subgroup of all p-subgroups of G.

My thoughts: for 1) maybe use sylow theorems. for 2) I don't understand because if N is a normal p-subgroup, then we have gNg^-1=N for all g. But by sylow, every p-subgroup is conjugate so we must conclude that N is the only p-subgroup of G.

1) By considering G/N, show that there is a subgroup H of G contaning N such that p does not divide [G:H].

2) Show that N is a subgroup of all p-subgroups of G.

My thoughts: for 1) maybe use sylow theorems. for 2) I don't understand because if N is a normal p-subgroup, then we have gNg^-1=N for all g. But by sylow, every p-subgroup is conjugate so we must conclude that N is the only p-subgroup of G.

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