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A subgroup H of finite index in a group G (finite or infinite) always contains a normal subgroup N (of G), also of finite index. In fact, if H has index n, then the index of N will be some divisor of n! and a multiple of n; indeed, N can be taken to be the kernel of the natural homomorphism from G to the permutation group of the left (or right ...
The index of a subgroup in a group [A 4 : H] = |A 4 |/|H| is the number of cosets generated by that subgroup. Since |A 4 | = 12 and |H| = 6, H will generate two left cosets, one that is equal to H and another, gH, that is of length 6 and includes all the elements in A 4 not in H. Since there are only 2 distinct cosets generated by H, then H ...
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.
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 ...
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.
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).
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.
In particular, if is the smallest prime dividing the order of , then every subgroup of index is normal. [ 21 ] The fact that normal subgroups of G {\displaystyle G} are precisely the kernels of group homomorphisms defined on G {\displaystyle G} accounts for some of the importance of normal subgroups; they are a way to internally classify all ...