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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.
Further, G is free on n = 2 generators, H has index e = [G : H] = 2 in G, and H is free on 1 + e(n–1) = 3 generators. The Nielsen–Schreier theorem states that like H, every subgroup of a free group can be generated as a free group, and if the index of H is finite, its rank is given by the index formula.
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 ...
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 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).
For a prime number, a Sylow p-subgroup (sometimes p-Sylow subgroup) of a group is a maximal-subgroup of , i.e., a subgroup of that is a p-group (meaning its cardinality is a power of ; or equivalently: For each group element, its order is some power of ) that is not a proper subgroup of any other -subgroup of .
In the mathematical field of group theory, the transfer defines, given a group G and a subgroup H of finite index, a group homomorphism from G to the abelianization of H.It can be used in conjunction with the Sylow theorems to obtain certain numerical results on the existence of finite simple groups.