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A subgroup H of a group G is called a characteristic subgroup if for every automorphism φ of G, one has φ(H) ≤ H; then write H char G. It would be equivalent to require the stronger condition φ(H) = H for every automorphism φ of G, because φ −1 (H) ≤ H implies the reverse inclusion H ≤ φ(H).
Suppose that G is a group, and H is a subset of G. For now, assume that the group operation of G is written multiplicatively, denoted by juxtaposition. Then H is a subgroup of G if and only if H is nonempty and closed under products and inverses. Closed under products means that for every a and b in H, the product ab is in H.
For example, if G is any non-trivial group, then the product G × G has a diagonal subgroup. Δ = { (g, g) : g ∈ G} which is not the direct product of two subgroups of G. The subgroups of direct products are described by Goursat's lemma. Other subgroups include fiber products of G and H.
A subset H of a group G is a subgroup of G if and only if it is nonempty and closed under products and inverses, that is, if and only if for every a and b in H, ab and a −1 are also in H. subgroup series A subgroup series of a group G is a sequence of subgroups of G such that each element in the series is a subgroup of the next element:
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.
In fact, C G (S) is always a normal subgroup of N G (S), being the kernel of the homomorphism N G (S) → Bij(S) and the group N G (S)/C G (S) acts by conjugation as a group of bijections on S. E.g. the Weyl group of a compact Lie group G with a torus T is defined as W(G,T) = N G (T)/C G (T), and especially if the torus is maximal (i.e. C G (T ...
A group G is called the direct sum [1] [2] of two subgroups H 1 and H 2 if each H 1 and H 2 are normal subgroups of G, the subgroups H 1 and H 2 have trivial intersection (i.e., having only the identity element of G in common), G = H 1, H 2 ; in other words, G is generated by the subgroups H 1 and H 2.
In general, a group homomorphism, : sends subgroups of to subgroups of . Also, the preimage of any subgroup of H {\displaystyle H} is a subgroup of G {\displaystyle G} . We call the preimage of the trivial group { e } {\displaystyle \{e\}} in H {\displaystyle H} the kernel of the homomorphism and denote it by ker f {\displaystyle \ker f} .