Subgroup generated by a subset

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Definition

Symbol-free definition

Given a group, and a subset of the group, the subgroup generated by that subset is defined in the following equivalent ways:

  • It is the intersection of all subgroups containing that subset
  • It is a subgroup containing that subset, such that the subset is a generating set for the subgroup

Definition with symbols

Given a group G and a subset S of G, the subgroup generated by S is defined in the following equivalent ways:

  • It is the intersection of all subgroups of G containing S
  • It is a subgroup H \le G such that S \subset H and S is a generating set for H

(Recall the fact that an intersection of subgroups is always a subgroup).

The subgroup generated by a subset S is denoted \langle S \rangle.

Examples

Extreme examples

  • If a subset S of a group G is a subgroup, then S equals the subgroup generated by S.
  • The subgroup generated by the empty subset is the trivial subgroup: it comprises only the identity element.

Other generic examples

  • The subgroup generated by a single element is the set of all its powers. In other words, the subgroup generated by an element g is the set of all elements expressible as g^n, n \in \mathbb{Z}. This is also termed the cyclic subgroup generated by g.

Examples in Abelian groups

  • In the group of integers under addition, the subgroup generated by the integers 4 and 6 is the subgroup of even integers.
  • In the group of rational numbers under addition, the subgroup generated by the rational number 1/2 is the group of integers and half-integers under addition.