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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 subgroup H contains only 0 and 4, and is isomorphic to /. There are four left cosets of H: H itself, 1+H, 2+H, and 3+H (written using additive notation since this is an additive group). Together they partition the entire group G into equal-size, non-overlapping sets. Thus the index [G : H] is 4.
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.
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 ...
The group G has the congruence subgroup property: a subgroup H has finite index in G if and only if there is a positive integer n such that (). The group G has solvable subgroup membership problem , that is, there is an algorithm that, given arbitrary words w , u 1 , ..., u n decides whether or not w represents an element of the 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).
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.
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 ...