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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. Centralizer and normalizer - Wikipedia

   en.wikipedia.org/wiki/Centralizer_and_normalizer

   If H is a subgroup of G, then N G (H) contains 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 ...

6. 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.

7. Lagrange's theorem (group theory) - Wikipedia

   en.wikipedia.org/wiki/Lagrange's_theorem_(group...

   Note that 3 is a factor of 6.) The number of such polynomials is the index in the symmetric group S n of the subgroup H of permutations that preserve the polynomial. (For the example of x + y − z, the subgroup H in S 3 contains the identity and the transposition (x y).) So the size of H divides n!. With the later development of abstract ...

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. 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.