Cosets and quotients · Definition

Coset

modern algebragroups

Before you start you need to know

Cosets cut a group into pieces the size of a subgroup, by translating the subgroup around the whole group.

Definition

The left coset of a subgroup \(H\) of a group \(G\) with representative \(a\in G\) is the set of products of \(a\) with the elements of \(H\).

\[aH=\{ah:\ h\in H\}\]

The coset \(aH\) is a translated copy of \(H\): the map \(h\mapsto ah\) is a bijection from \(H\) onto \(aH\), so every coset has \(|H|\) elements. In additive notation we write \(a+H\); for example, for \(H=\{0,3,6,9\}\) in \(\mathbb{Z}_{12}\), the coset \(1+H\) is \(\{1,4,7,10\}\).

We have \(aH=bH\) if and only if \(a^{-1}b\in H\), and two cosets are either equal or disjoint. Each element \(a\) lies in its own coset \(aH\) (since \(e\in H\)), so the cosets form a partition of \(G\): the one given by the equivalence relation \(a\sim b\iff a^{-1}b\in H\). The coset of \(e\) is \(H\) itself.

There are also right cosets \(Ha\). In an abelian group they agree with the left ones, but not in general: in \(S_3\), with \(H=\{e,(1\,2)\}\), the cosets \((1\,3)H\) and \(H(1\,3)\) are different. This difference is what later separates normal subgroups from the rest.

Splitting \(G\) into cosets of equal size is the idea behind Lagrange's theorem: if \(G\) is finite, the number of cosets times \(|H|\) gives \(|G|\). Beware of confusing the representative with the coset: many different elements \(a\) produce the same set \(aH\).

Think about it

How many distinct cosets does \(H=\{0,5,10,15\}\) have in \(\mathbb{Z}_{20}\)?

Each coset has \(4\) elements and they split the \(20\): there are \(5\) cosets.

Why is \(aH=H\) when \(a\in H\)?

Because \(H\) is closed under products: \(ah\in H\), and every \(h\in H\) arises as \(a(a^{-1}h)\).

Exercises 10

  1. Let \(H=\{0,3,6,9\}\) in \((\mathbb{Z}_{12},+)\). What is the coset \(1+H\)?
  2. Let \(H=\{0,4,8\}\) in \((\mathbb{Z}_{12},+)\). How many distinct cosets does \(H\) have?
  3. In \((\mathbb{Z},+)\) let \(H=5\mathbb{Z}\). Which coset equals \(17+H\)?
  4. Let \(H\) be a subgroup of order \(4\) of \((\mathbb{Z}_{20},+)\). How many elements does the coset \(7+H\) have?
  5. Let \(H=\{0,3,6,9\}\) in \((\mathbb{Z}_{12},+)\). Which of these elements lies in the same coset as \(5\)?
  6. In the multiplicative group \(\mathbb{Z}_7^{*}\) let \(H=\{1,6\}\). What is the coset \(3H\)?
  7. Two left cosets \(aH\) and \(bH\) share an element. What follows?
  8. Let \(H=\langle 12\rangle\) in \((\mathbb{Z}_{36},+)\). How many distinct cosets are there?
  9. In \(S_3\) let \(H=\{e,(1\,2)\}\) and \(g=(1\,3)\). How are \(gH\) and \(Hg\) related?
  10. Let \(H=\langle 6\rangle\) in \((\mathbb{Z}_{18},+)\). Which coset equals \(5+H\)?

References

  • David S. Dummit, Richard M. Foote, Abstract Algebra, 3rd edition, Wiley, 2004. §3.1
  • Joseph A. Gallian, Contemporary Abstract Algebra, 9th edition, Cengage, 2017. cap. 7
  • I. N. Herstein, Abstract Algebra, 3rd edition, Wiley, 1996. §2.4
  • John B. Fraleigh, A First Course in Abstract Algebra, 7th edition, Pearson, 2002. §10