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The index of the normal subgroup not only has to be a divisor of n!, but must satisfy other criteria as well. 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.
The set G is open and a topological group. Consider the identity component. G 0, or in other words the connected component containing the identity 1 of A; G 0 is a normal subgroup of G. The quotient group. Λ A = G/G 0. is the abstract index group of A. Because G 0, being the component of an open set, is both open and closed in G, the index ...
This is the group obtained from the orthogonal group in dimension 2n + 1 by taking the kernel of the determinant and spinor norm maps. B 1 (q) also exists, but is the same as A 1 (q). B 2 (q) has a non-trivial graph automorphism when q is a power of 2. This group is obtained from the symplectic group in 2n dimensions by quotienting out the center.
A consequence of the theorem is that the order of any element a of a finite group (i.e. the smallest positive integer number k with a k = e, where e is the identity element of the group) divides the order of that group, since the order of a is equal to the order of the cyclic subgroup generated by a. If the group has n elements, it follows
For example, in the symmetric group shown above, where ord(S 3) = 6, the possible orders of the elements are 1, 2, 3 or 6. The following partial converse is true for finite groups: if d divides the order of a group G and d is a prime number, then there exists an element of order d in G (this is sometimes called Cauchy's theorem).
In group theory, a maximal subgroup H of a group G is a proper subgroup, such that no proper subgroup K contains H strictly. In other words, H is a maximal element of the partially ordered set of subgroups of G that are not equal to G .
For n > 1, the group A n is the commutator subgroup of the symmetric group S n with index 2 and has therefore n!/2 elements. It is the kernel of the signature group homomorphism sgn : S n → {1, −1} explained under symmetric group. The group A n is abelian if and only if n ≤ 3 and simple if and only if n = 3 or n ≥ 5.
The group consisting of all permutations of a set M is the symmetric group of M. p-group If p is a prime number, then a p-group is one in which the order of every element is a power of p. A finite group is a p-group if and only if the order of the group is a power of p. p-subgroup A subgroup that is also a p-group.