Left cosets are in bijection via left multiplication

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This article gives the statement, and possibly proof, of a basic fact in group theory.
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Statement

Statement with symbols

Let H be a subgroup of a group G and let xH and yH be two left cosets of H. Then, there is a bijection between xH and yH as subsets of G, given by the left multiplication by yx^{-1}.


Related facts

Similar facts

Related facts about other algebraic structures

Significance

In the particular case where the subgroup is also normal, the cosets of the subgroup are congruence classes for the congruence corresponding to the normal subgroup. The result then tells us that all the congruence classes have equal size for any congruence, which tells us that the variety of groups is a congruence-uniform variety (viz, every congruence on it is a uniform congruence). Further information: variety of groups is congruence-uniform


Facts used

  1. Invertible implies cancellative in monoid: In particular, we can cancel elements in a group: if ab = ac, then b = c.

Proof

Given: A group G, a subgroup H, and two left cosets xH, yH of H

To prove: Left multiplication by yx^{-1} establishes a bijection between xH and yH.

Proof: We prove that left multiplication by yx^{-1} sends xH to yH, is surjective, and is injective.

  1. Well-defined as a map from xH to yH: First, note that if g = xh then yx^{-1}g = yh. Thus, any element in xH gets mapped to an element in yH.
  2. Surjective: Every element of the form yh with h in H arises as yx^{-1}(xh), hence, it arises as the image of left multiplication by yx^{-1}. Thus, the map from xH to yH is surjective.
  3. Injective: Given two distinct elements xh_1, xh_2 \in xH, the elements (yx^{-1})xh_1 = yh_1 and (yx^{-1})xh_2 = yh_2 are also distinct, because by fact (1), if they were equal, then canceling yx^{-1} from both sides would give xh_1 = xh_2. Thus, left multiplication by yx^{-1} sends distinct elements to distinct elements, so the map is injective.

Thus, left multiplication by yx^{-1} is a bijection from xH to yH.

Sidenote

In general, there is no natural bijection between two left cosets -- the bijection depends on a choice of element in both cosets (the elements x and y in the above description).



References

  • Algebra by Michael Artin, ISBN 0130047635, 13-digit ISBN 978-0130047632, More info, Page 57, in text between points (6.8) and (6.9). Full justification is not provided