Difference between revisions of "Left cosets partition a group"

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Statement

Verbal statement

The following equivalent statements are true:

1. The left cosets of a subgroup in a group partition the group.
2. The relation of one element being in the left coset of the other, is an equivalence relation.
3. Every element of the group is in exactly one left coset.
4. 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:

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

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 $gH$, 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).

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.

Proof in form (2)

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

To prove: The relation $a \sim b \iff \ \exists \ h \in H$ such that $a = bh$, 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 $a \in G$, $a = ae$, 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 $a = bh$, for some $h \in H$, then $b = ah^{-1}$. 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, $a \sim b \implies b \sim a$.

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 $a = bh$, and $b = ck$, for $h, k \in H$, and $a = ckh$. Since $H$ is a subgroup, $h,k \in H \implies kh \in H$, 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 $aH$ and $bH$ are either equal or disjoint (they have empty intersection)

Proof: We'll assume that $aH$ and $bH$ are not disjoint, and prove that they are equal.

For this, suppose $c \in aH \cap bH$. Then, there exist $h_1,h_2$ such that $ah_1 = bh_2 = c$. Thus, $b = ah_1h_2^{-1} \in aH$ and $a = bh_2h_1^{-1} \in bH$.

Now, for any element $ah \in aH$, we have $ah = bh_2h_1^{-1}h \in bH$, and similarly, for every element $bh \in bH$, we have $bh = ah_1h_2^{-1}h \in aH$. Thus, $aH \subseteq bH$ and $bH \subseteq aH$, so $aH = bH$.

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:

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

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

References

Textbook references

• Abstract Algebra by David S. Dummit and Richard M. Foote, 10-digit ISBN 0471433349, 13-digit ISBN 978-0471433347, More info, Proposition 4, Page 80