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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 ...
A metacyclic group is a group containing a cyclic normal subgroup whose quotient is also cyclic. [23] These groups include the cyclic groups, the dicyclic groups, and the direct products of two cyclic groups. The polycyclic groups generalize metacyclic groups by allowing more than one level of group extension. A group is polycyclic if it has a ...
Cycles that contain a non-prime number of elements have cyclic subgroups that are not shown in the graph. For the group Dih 4 above, we could draw a line between a 2 and e since (a 2) 2 = e, but since a 2 is part of a larger cycle, this is not an edge of the cycle graph. There can be ambiguity when two cycles share a non-identity element.
A subgroup H of a group G is ascendant if there is an ascending subgroup series starting from H and ending at G, such that every term in the series is a normal subgroup of its successor. The series may be infinite. If the series is finite, then the subgroup is subnormal. automorphism An automorphism of a group is an isomorphism of the group to ...
G 0 is the trivial subgroup; G i is a normal subgroup of G i+1 (for every i between 0 and n - 1) and the quotient group G i+1 / G i is a cyclic group (for every i between 0 and n - 1) A metacyclic group is a polycyclic group with n ≤ 2, or in other words an extension of a cyclic group by a cyclic group.
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.
The additive group of rational numbers (Q, +) is locally cyclic – any pair of rational numbers a/b and c/d is contained in the cyclic subgroup generated by 1/(bd). [2]The additive group of the dyadic rational numbers, the rational numbers of the form a/2 b, is also locally cyclic – any pair of dyadic rational numbers a/2 b and c/2 d is contained in the cyclic subgroup generated by 1/2 max ...
Bounded generation is unaffected by passing to a subgroup of finite index: if H is a finite index subgroup of G then G is boundedly generated if and only if H is boundedly generated. Bounded generation goes to extension: if a group G has a normal subgroup N such that both N and G/N are boundedly generated, then so is G itself.