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A normal subgroup of a normal subgroup of a group need not be normal in the group. That is, normality is not a transitive relation. The smallest group exhibiting this phenomenon is the dihedral group of order 8. [15] However, a characteristic subgroup of a normal subgroup is normal. [16] A group in which normality is transitive is called a T ...
Due to the maximality condition, if is any -subgroup of , then is a subgroup of a -subgroup of order . An important consequence of Theorem 2 is that the condition n p = 1 {\displaystyle n_{p}=1} is equivalent to the condition that the Sylow p {\displaystyle p} -subgroup of G {\displaystyle G} is a normal subgroup (Theorem 3 can often show n p ...
The proof of Clifford's theorem is best explained in terms of modules ... There is an Abelian normal subgroup N of order 4 (a Klein 4-subgroup) ...
The field E H is a normal extension of F (or, equivalently, Galois extension, since any subextension of a separable extension is separable) if and only if H is a normal subgroup of Gal(E/F). In this case, the restriction of the elements of Gal( E / F ) to E H induces an isomorphism between Gal( E H / F ) and the quotient group Gal( E / F )/ H .
Diagram of the fundamental theorem on homomorphisms, where is a homomorphism, is a normal subgroup of and is the identity element of .. Given two groups and and a group homomorphism:, let be a normal subgroup in and the natural surjective homomorphism / (where / is the quotient group of by ).
Applying the Schur–Zassenhaus theorem to G/A reduces the proof to the case when H=A is abelian which has been done in the previous step. If the normalizer N=N G (P) of every p-Sylow subgroup P of H is equal to G, then H is nilpotent, and in particular solvable, so the theorem follows by the previous step.
Frattini's argument can be used as part of a proof that any finite nilpotent group is a direct product of its Sylow subgroups.; By applying Frattini's argument to (()), it can be shown that (()) = whenever is a finite group and is a Sylow -subgroup of .
For example, the subgroup Z 7 of the non-abelian group of order 21 is normal (see List of small non-abelian groups and Frobenius group#Examples). An alternative proof of the result that a subgroup of index lowest prime p is normal, and other properties of subgroups of prime index are given in .