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A cyclic group is a group which is equal to one of its cyclic subgroups: G = g for some element g, called a generator of G. For a finite cyclic group G of order n we have G = {e, g, g 2, ... , g n−1}, where e is the identity element and g i = g j whenever i ≡ j (mod n); in particular g n = g 0 = e, and g −1 = g n−1.
The lattice of subgroups of the infinite cyclic group can be described in the same way, as the dual of the divisibility lattice of all positive integers. If the infinite cyclic group is represented as the additive group on the integers, then the subgroup generated by d is a subgroup of the subgroup generated by e if and only if e is a divisor ...
The Schur multipliers of the alternating groups A n (in the case where n is at least 4) are the cyclic groups of order 2, except in the case where n is either 6 or 7, in which case there is also a triple cover. In these cases, then, the Schur multiplier is the cyclic group of order 6, and the covering group is a 6-fold cover. H 2 (A n, Z) = 0 ...
To see this, given a group G, consider the free group F G on G. By the universal property of free groups, there exists a unique group homomorphism φ : F G → G whose restriction to G is the identity map. Let K be the kernel of this homomorphism. Then K is normal in F G, therefore is equal to its normal closure, so G | K = F G /K.
In group theory, especially, in geometric group theory, the class of free-by-cyclic groups have been deeply studied as important examples. A group G {\displaystyle G} is said to be free-by-cyclic if it has a free normal subgroup F {\displaystyle F} such that the quotient group G / F {\displaystyle G/F} is cyclic .
In the quaternion group of order 8, each of the cyclic subgroups of order 4 is normal, but none of these are characteristic. However, the subgroup, {1, −1}, is characteristic, since it is the only subgroup of order 2. If n > 2 is even, the dihedral group of order 2n has 3 subgroups of index 2, all of which are normal. One of these is the ...
A 3-group with two generators is regular if and only if its derived subgroup is cyclic. Every p-group of odd order with cyclic derived subgroup is regular. The subgroup of a p-group G generated by the elements of order dividing p k is denoted Ω k (G) and regular groups are well-behaved in that Ω k (G) is precisely the set of elements of order ...
The automorphism group of the graph includes a cyclic subgroup that acts transitively on the graph's vertices. In other words, the graph has an automorphism which is a cyclic permutation of its vertices. The graph has an adjacency matrix that is a circulant matrix.