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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 subgroup of H that is invariant under all inner automorphisms is called normal; also, an invariant subgroup. ∀φ ∈ Inn(G)： φ(H) ≤ H. Since Inn(G) ⊆ Aut(G) and a characteristic subgroup is invariant under all automorphisms, every characteristic subgroup is normal. However, not every normal subgroup is characteristic.
There is no requirement made that A i be a normal subgroup of G, only a normal subgroup of A i +1. The quotient groups A i +1 /A i are called the factor groups of the series. If in addition each A i is normal in G, then the series is called a normal series, when this term is not used for the weaker sense, or an invariant series.
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.
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.
The first term, S 1 (G), is the subgroup generated by the minimal normal subgroups and so is equal to the socle of G. For this reason the upper exponent- p central series is sometimes known as the socle series or even the Loewy series, though the latter is usually used to indicate a descending series.
The normal core of a subgroup H of a group G is the largest normal subgroup of G that is contained in H. normal series A normal series of a group G is a sequence of normal subgroups of G such that each element of the sequence is a normal subgroup of the next element:
More precisely, H is a subgroup of G if the restriction of ∗ to H × H is a group operation on H. This is often denoted H ≤ G, read as "H is a subgroup of G". The trivial subgroup of any group is the subgroup {e} consisting of just the identity element. [1] A proper subgroup of a group G is a subgroup H which is a proper subset of G (that ...