# Tour:Left cosets are in bijection via left multiplication

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**This article adapts material from the main article:** left cosets are in bijection via left multiplication

This page is part of the Groupprops guided tour for beginners (Jump to beginning of tour)PREVIOUS: Left cosets partition a group|UP: Introduction three (beginners)|NEXT: Right coset of a subgroup

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The different left cosets of a subgroup are in bijection with each other, under the action of the group via left multiplication. This means that all left cosets of a subgroup are of the same size.PREREQUISITES: Definition of group, subgroup and coset, invertible implies cancellative, and Tour:Mind's eye test two (beginners)#Left and right multiplication mapsWHAT YOU NEED TO DO: Read and understand the statement and proof below. If you find it hard, refer back to the suggested prerequisites pages.

## Statement

### Statement with symbols

Let be a subgroup of a group and let and be two left cosets of . Then, there is a bijection between and as subsets of , given by the *left multiplication* by .

## Facts used

- Invertible implies cancellative in monoid: In particular, we can cancel elements in a group: if , then .

## Proof

**Given**: A group , a subgroup , and two left cosets , of

**To prove**: Left multiplication by establishes a bijection between and .

**Proof**: We prove that left multiplication by sends to , is surjective, and is injective.

- Well-defined as a map from to : First, note that if then . Thus, any element in gets mapped to an element in .
- Surjective: Every element of the form with in arises as , hence, it arises as the image of left multiplication by . Thus, the map from to is surjective.
- Injective: Given two distinct elements , the elements and are also distinct, because by fact (1), if they were equal, then canceling from both sides would give . Thus, left multiplication by sends distinct elements to distinct elements, so the map is injective.

Thus, left multiplication by is a bijection from to .

### 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 and in the above description).

This page is part of the Groupprops guided tour for beginners (Jump to beginning of tour)PREVIOUS: Left cosets partition a group|UP: Introduction three (beginners)|NEXT: Right coset of a subgroup

General instructions for the tour | Pedagogical notes for the tour | Pedagogical notes for this part