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2. Index of a subgroup - Wikipedia

   en.wikipedia.org/wiki/Index_of_a_subgroup

   Since the normal subgroup is a subgroup of H, its index in G must be n times its index inside H. Its index in G must also correspond to a subgroup of the symmetric group S n, the group of permutations of n objects. So for example if n is 5, the index cannot be 15 even though this divides 5!, because there is no subgroup of order 15 in S 5.

3. Glossary of group theory - Wikipedia

   en.wikipedia.org/wiki/Glossary_of_group_theory

   A subgroup of a group G is a subset H of the elements of G that itself forms a group when equipped with the restriction of the group operation of G to H × 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 ...

4. Characteristic subgroup - Wikipedia

   en.wikipedia.org/wiki/Characteristic_subgroup

   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).

5. Subgroup - Wikipedia

   en.wikipedia.org/wiki/Subgroup

   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.

6. Nielsen–Schreier theorem - Wikipedia

   en.wikipedia.org/wiki/Nielsen–Schreier_theorem

   The free group G = π 1 (X) has n = 2 generators corresponding to loops a,b from the base point P in X.The subgroup H of even-length words, with index e = [G : H] = 2, corresponds to the covering graph Y with two vertices corresponding to the cosets H and H' = aH = bH = a −1 H = b − 1 H, and two lifted edges for each of the original loop-edges a,b.

7. Three subgroups lemma - Wikipedia

   en.wikipedia.org/wiki/Three_subgroups_lemma

   If H and K are subgroups of a group G, the commutator of H and K, denoted by [H, K], is defined as the subgroup of G generated by commutators between elements in the two subgroups. If L is a third subgroup, the convention that [H,K,L] = [[H,K],L] will be followed.

8. Core (group theory) - Wikipedia

   en.wikipedia.org/wiki/Core_(group_theory)

   For a group G, the normal core or normal interior [1] of a subgroup H is the largest normal subgroup of G that is contained in H (or equivalently, the intersection of the conjugates of H). More generally, the core of H with respect to a subset S ⊆ G is the intersection of the conjugates of H under S, i.e.

9. Restricted representation - Wikipedia

   en.wikipedia.org/wiki/Restricted_representation

   The special case where σ is the trivial representation of H was first used extensively by Hua in his work on the Szegő kernels of bounded symmetric domains in several complex variables, where the Shilov boundary has the form G/H. [4] [5] More generally the Cartan-Helgason theorem gives the decomposition when G/H is a compact symmetric space ...