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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 ...
Since this is true for any d, x must be a member of A, so ca = xc implies that cac −1 ∈ A and therefore A is a normal subgroup. The index of the normal subgroup not only has to be a divisor of n!, but must satisfy other criteria as well. Since the normal subgroup is a subgroup of H, its index in G must be n times its index inside H.
If H is a subgroup of G, then the largest subgroup of G in which H is normal is the subgroup N G (H). If S is a subset of G such that all elements of S commute with each other, then the largest subgroup of G whose center contains S is the subgroup C G (S). A subgroup H of a group G is called a self-normalizing subgroup of G if N G (H) = H.
If A and B are normal, then A × B is a normal subgroup of G × H. Moreover, the quotient of the direct products is isomorphic to the direct product of the quotients: (G × H) / (A × B) ≅ (G / A) × (H / B). Note that it is not true in general that every subgroup of G × H is the product of a subgroup of G with a subgroup of H.
That is, we let R be the subgroup generated by the strings rfrf, r 8, f 2, each of which is also equivalent to 1 when considered as products in D 8. If we then let N be the subgroup of F generated by all conjugates x −1 Rx of R, then it follows by definition that every element of N is a finite product x 1 −1 r 1 x 1... x m −1 r m x m ...
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 ...
If G is the semidirect product of the normal subgroup N and the subgroup H, and both N and H are finite, then the order of G equals the product of the orders of N and H. This follows from the fact that G is of the same order as the outer semidirect product of N and H, whose underlying set is the Cartesian product N × H.
In the context of group theory, the socle of a group G, denoted soc(G), is the subgroup generated by the minimal normal subgroups of G.It can happen that a group has no minimal non-trivial normal subgroup (that is, every non-trivial normal subgroup properly contains another such subgroup) and in that case the socle is defined to be the subgroup generated by the identity.