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If H is a subgroup of G, the index of the normal core of H satisfies the following inequality: |: | |: |! where ! denotes the factorial function; this is discussed further below. As a corollary, if the index of H in G is 2, or for a finite group the lowest prime p that divides the order of G, then H is normal, as the index of its core must ...
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 ...
Burnside's fusion theorem can be used to give a more powerful factorization called a semidirect product: if G is a finite group whose Sylow p-subgroup P is contained in the center of its normalizer, then G has a normal subgroup K of order coprime to P, G = PK and P∩K = {1}, that is, G is p-nilpotent.
The table below lists the largest currently known prime numbers and probable primes (PRPs) as tracked by the PrimePages and by Henri & Renaud Lifchitz's PRP Records. Numbers with more than 2,000,000 digits are shown.
G is the group /, the integers mod 8 under addition. 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).
For any integer k, we have a k = e if and only if ord(a) divides k. In general, the order of any subgroup of G divides the order of G. More precisely: if H is a subgroup of G, then ord(G) / ord(H) = [G : H], where [G : H] is called the index of H in G, an integer. This is Lagrange's theorem. (This is, however, only true when G has finite order.
In mathematics, specifically group theory, Cauchy's theorem states that if G is a finite group and p is a prime number dividing the order of G (the number of elements in G), then G contains an element of order p. That is, there is x in G such that p is the smallest positive integer with x p = e, where e is the identity element of G.
If K is a complement of H in G then K forms both a left and right transversal of H. That is, the elements of K form a complete set of representatives of both the left and right cosets of H. The Schur–Zassenhaus theorem guarantees the existence of complements of normal Hall subgroups of finite groups.