Groups and Group Actions: Lecture 7

In which we think about the link between equivalence relations and partitions, and meet cosets.

  • Definition of an equivalence class.
  • Definition of a partition of a set.
  • Theorem 27: Let \sim be an equivalence relation on a set S.  The equivalence classes of \sim partition S.  To prove this, we checked that each equivalence class is non-empty, that between them they cover the set (their union is the set), and they are pairwise disjoint, using each of reflexivity, symmetry and transitivity along the way.
  • Theorem 28: Let P be a partition of a set S.  For a \in S, write P_a for the unique part in P with a \in P_a.  Define a relation \sim on S by a \sim b if and only if b \in P_a.  Then \sim is an equivalence relation.  We checked reflexivity, symmetry and transitivity.
  • Corollary 29: There is a bijection between equivalence relations on a set S and partitions of the same set $latex $.  This was a consequence of Theorems 27 and 28.
  • Definition of \mathbb{Z}_n, and of binary operations + and \times on \mathbb{Z}_n.
  • Lemma 30: The operations + and \times on \mathbb{Z}_n are well defined.  Once we’d worked out what we needed to worry about, it was straightforward to check that in fact we didn’t need to worry.
  • Proposition 31: (\mathbb{Z}_n, +) is an Abelian group.  Moreover, it is cyclic and isomorphic to C_n.  Furthermore, \times is associative and commutative on \mathbb{Z}_n, and \times is distributive over +.  The proof of this is an exercise.
  • Proposition 32:
    1.  Take \overline{x} \in \mathbb{Z}_n.  Then \overline{x} has a multiplicative inverse in \mathbb{Z}_n if and only if \mathrm{hcf}(x,n)=1.
    2. If p is prime, then \mathbb{Z}_p is a field.
    3. Let \mathbb{Z}_n^{\times} = \{ \overline{x} \in \mathbb{Z}_n : \overline{x} has a multiplicative inverse \} be the set of units in \mathbb{Z}_n. Then \mathbb{Z}_n^{\times} is a group under multiplication.  The proof is an exercise on Sheet 4.
  • Definition of a left coset of a subgroup H in a group G.  Definition of the notation G / H for the set of cosets.  Definition of the index of a subgroup in a group.  Definition of right cosets.

Understanding today’s lecture

Can you prove Propositions 31 and 32?

What are the left cosets of A_n in S_n?  What is the index of A_n in S_n?

Pick a group and a subgroup.  What are the left cosets?  For example, you could pick a small dihedral group like D_8 (the symmetries of a square) and explore left cosets of subgroups of D_8.

Further reading

In Lemma 30, we showed that two functions we’d just written down were well defined.  What does well defined mean?  Here are some useful thoughts from Tim Gowers.

I’ve written before about the result that the non-zero integers modulo a prime form a group under multiplication, which is closely related to \mathbb{Z}_p (for p prime) being a field.

And here’s something a bit lighter, in case you’re feeling old as the end of term approaches.

Preparation for Lecture 8

Can you say anything interesting about the index of a subgroup H in a finite group G, in terms of the orders of G and H?

Take a finite group G, and a subgroup H.  Can you show that the left cosets of H partition G?  What is the corresponding equivalence relation on the set G?

Can you show that all the left cosets of H in G have the same size?

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