Tour:Left cosets partition a group

This article adapts material from the main article: left cosets partition a group

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WHAT YOU NEED TO DO:
Read the statements below, and understand why they are equivalent. You may wish to do some background reading on equivalence relations and partitions if the equivalence between the statements is not clear.

Make sure you understand the proof. If you were able to prove the equivalence of definitions of left coset earlier on, this proof should not be hard to understand -- it is essentially the same thing.

Statement

Verbal statement

The following equivalent statements are true:

The left cosets of a subgroup in a group partition the group.
The relation of one element being in the left coset of the other, is an equivalence relation.
Every element of the group is in exactly one left coset.
Any two left cosets of a subgroup either do not intersect, or are equal.

Statement with symbols

Suppose $G$ is a group, and $H$ is a subgroup. Then, the following equivalent statements are true:

The left cosets of $H$ , namely $g H, g \in G$ , form a partition of the group $G$ . In other words, $G$ is a disjoint union of left cosets of $H$ .
The relation $a \sim b ⟺ a \in b H$ is an equivalence relation on $G$
For every $g \in G$ , there is exactly one left coset of $H$ in $G$ containing $g$ .
If $a H$ and $b H$ are left cosets of $H$ in $G$ , then either $a H = b H$ or $a H \cap b H$ is empty.

Equivalence of statements

These statements are equivalent because of the following general fact about sets and equivalence relations. If $S$ is a set, and $\sim$ is an equivalence relation on $S$ , then we can partition $S$ as a disjoint union of equivalence classes under $\sim$ . Two elements $a$ and $b$ are defined to be in the same equivalence class under $\sim$ if $a \sim b$ .

Conversely, if $S$ is partitioned as a disjoint union of subsets, then the relation of being in the same subset is an equivalence relation on $S$ .

Hence, there is a correspondence between equivalence relations on a set and partitions of the set into subsets. This statement about left cosets thus states that the left cosets partition the group, which is also the same as saying that the relation of one element being in the left coset of another, is an equivalence relation.

Here, we give the proof both in form (2) and form (4). The two proofs are essentially the same, but they are worked out in somewhat different language, and explain how to think both in terms of equivalence relations and in terms of partitions.

Definitions used

Let $G$ be a group, $H$ be a subgroup.

For $a, b \in G$ , we say that $a$ is in the left coset of $b$ with respect to $H$ if there exists $h \in H$ such that $a = b h$ .

Proof in form (2)

Given: A group $G$ , a subgroup $H$

To prove: The relation $a \sim b ⟺ \exists h \in H$ such that $a = b h$ , is an equivalence relation on $G$

Reflexivity

To prove: For any $a \in G$ , $a \sim a$ .

Proof: Clearly $e \in H$ (since $H$ is a subgroup). Hence, for any $a \in G$ , $a = a e$ , so $a \sim a$ : $a$ is in its own left coset.

Symmetry

To prove: For any $a, b \in G$ such that $a \sim b$ , we have $b \sim a$ .

Proof: If $a = b h$ , for some $h \in H$ , then $b = a h^{- 1}$ . Since $h \in H$ and $H$ is a subgroup, $h^{- 1} \in H$ . Thus, if $a$ is in the left coset of $b$ , then $b$ is in the left coset of $a$ . In symbols, $a \sim b ⟹ b \sim a$ .

Transitivity

To prove: If $a, b, c \in G$ are such that $a \sim b$ , and $b \sim c$ , then $a \sim c$

Proof: If $a = b h$ , and $b = c k$ , for $h, k \in H$ , and $a = c k h$ . Since $H$ is a subgroup, $h, k \in H ⟹ k h \in H$ , so $a$ is in the left coset of $c$ .

Proof in form (4)

Given: A group $G$ , a subgroup $H$ , two elements $a, b \in G$

To prove: The left cosets $a H$ and $b H$ are either equal or disjoint (they have empty intersection)

Proof: We'll assume that $a H$ and $b H$ are not disjoint, and prove that they are equal.

For this, suppose $c \in a H \cap b H$ . Then, there exist $h_{1}, h_{2}$ such that $a h_{1} = b h_{2} = c$ . Thus, $b = a h_{1} h_{2}^{- 1} \in a H$ and $a = b h_{2} h_{1}^{- 1} \in b H$ .

Now, for any element $a h \in a H$ , we have $a h = b h_{2} h_{1}^{- 1} h \in b H$ , and similarly, for every element $b h \in b H$ , we have $b h = a h_{1} h_{2}^{- 1} h \in a H$ . Thus, $a H \subseteq b H$ and $b H \subseteq a H$ , so $a H = b H$ .

WHAT'S MORE: Some alternative approaches to understanding the proof. Ignore those that use unfamiliar terminology.

Related facts

Converse

A partial converse to this result is true. If $H$ is a subset of $G$ containing the identity element with the property that the set of all left translates of $H$ , i.e. the set of subsets $g H$ , form a partition of $G$ , then $H$ is a subgroup of $G$ .

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

Analogues in other algebraic structures

The proof that the left cosets of a subgroup partition the group uses all the properties of groups: the existence of identity element is used to prove reflexivity, the existence of inverses (along with associativity and the identity element) was used to prove symmetry, and associativity is used to prove transitivity. Hence, extending the result to algebraic structures weaker than groups is in general hard. There are, however, some ways of extending.

Statement	Analogue of group	Analogue of subgroup	Comment
Left cosets of a subgroup partition a monoid	monoid (associative, identity element, not necessarily any inverses)	submonoid that is in fact a group	We do not require the bigger structure to be a group. All we need is associativity in the bigger structure. Thus, the left cosets of a subgroup in a monoid, still partition it. (Note that we still do require associativity in the bigger structure).

Relation with right cosets and normal subgroups

Right cosets partition a group: The proof of this is analogous to that for left cosets.
Left cosets are in bijection via left multiplication: In particular, any two left cosets of a subgroup have the same size as the subgroup.
Equivalence of definitions of coset: A subset of a group occurs as the left coset of a subgroup if and only if it occurs as the right coset of a subgroup.
For a normal subgroup, the set of left cosets coincides with the set of right cosets, and this set can be given the structure of a group called the quotient group. Any homomorphism of groups is obtained by composing the quotient map to a quotient group with an injection: see normal subgroup equals kernel of homomorphism and first isomorphism theorem.

Other related facts

Lattice of subgroups embeds in partition lattice: As we see here, every subgroup gives rise to a partition of the group (namely, the partition into left cosets). This gives a function from the lattice of subgroups of a group to the partition lattice of the group. It turns out that this map is a lattice embedding, i.e., it preserves the lattice operations.

Other proofs

Orbits under a group action

One easy way of seeing that the left cosets partition a group is by viewing the left cosets as orbits of the group under the action of the subgroup by right multiplication.

Left congruence

Another way of viewing the partition of a group into left cosets of a subgroup is in terms of a left congruence. A left congruence on a magma $(S, *)$ is an equivalence relation $\sim$ with the property that:

$a \sim b ⟹ c * a \sim c * b \forall c \in S$

The only left congruences on a group are those that arise as partitions in terms of left cosets of a subgroup.

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