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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 ...
A proper subgroup of a group G is a subgroup H which is a proper subset of G (that is, H ≠ G). This is often represented notationally by H < G, read as "H is a proper subgroup of G". Some authors also exclude the trivial group from being proper (that is, H ≠ {e} ). [2] [3] If H is a subgroup of G, then G is sometimes called an overgroup of H.
normal subgroup A subgroup N of a group G is normal in G (denoted N G) if the conjugation of an element n of N by an element g of G is always in N, that is, if for all g ∈ G and n ∈ N, gng −1 ∈ N. A normal subgroup N of a group G can be used to construct the quotient group G / N. normalizer
One way of thinking about the N ⋊ H symbol is as a combination of the symbol for normal subgroup ( ) and the symbol for the product (×). Barry Simon , in his book on group representation theory, [ 12 ] employs the unusual notation N Ⓢ φ H {\displaystyle N\mathbin {\circledS _{\varphi }} H} for the semidirect product.
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.
O p (G) is the intersection of all normal subgroups K of G such that G/K is a (possibly non-abelian) p-group (i.e., K is an index normal subgroup): G/O p (G) is the largest p-group (not necessarily abelian) onto which G surjects. O p (G) is also known as the p-residual subgroup.
A core-free subgroup is a subgroup whose normal core is the trivial subgroup. Equivalently, it is a subgroup that occurs as the isotropy subgroup of a transitive, faithful group action. The solution for the hidden subgroup problem in the abelian case generalizes to finding the normal core in case of subgroups of arbitrary groups.
Similarly, a normal subgroup N of G is said to be a maximal normal subgroup (or maximal proper normal subgroup) of G if N < G and there is no normal subgroup K of G such that N < K < G. We have the following theorem: Theorem: A normal subgroup N of a group G is a maximal normal subgroup if and only if the quotient G/N is simple.