Subset containing identity whose left translates partition the group is a subgroup

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This article gives a proof/explanation of the equivalence of multiple definitions for the term subgroup
View a complete list of pages giving proofs of equivalence of definitions

Statement

Suppose G is a group and H is a nonempty subset of G containing the identity element of G. Then, the following are equivalent:

  1. H is a subgroup of G
  2. For any x,y \in H, either xH = yH or xH \cap yH is empty
  3. For any g \in G, either H = gH or H \cap gH is empty

Related facts

A reformulation, or immediate corollary, of this is the statement that:

Left congruence on a group equals left coset space relation from a subgroup

Definitions used

Subgroup

We shall use the definition of subgroup in terms of the subgroup criterion. A nonempty subset H of a group G is termed a subgroup if for any a,b \in H, the element a^{-1}b is also in H.

Proof

(1) implies (2) implies (3)

This is a direct consequence of the fact that for any subgroup, the left cosets partition the group.

(3) implies (1)

Given: A group G and nonempty subset H such that for every g \in G either H = gH or H \cap gH is empty.

To prove: For any a,b \in H, the element a^{-1}b is also in H

Proof: Let a,b \in H. Consider the set aH. Since H contains the identity element, a \in aH, so aH \cap H is nonempty. Thus, by assumption, aH = H, so there exists h \in H such that ah = b, so h = a^{-1}b. Thus, a^{-1}b is in H, completing the proof.