Define the relation \(\sim\) on \(\Z\) by \(a \sim b\) if and only if \(5 \mid b-a\text{.}\) Prove that \(\sim\) is an equivalence relation and describe the equivalence classes.
Section Week 9 (10/20-10/25)
Handout Monday 10/20
Last time we explored the concepts of partitions and equivalence relations. Let’s look at an example of this.
Example 1.
The next learning target asks you to prove that a relation is an equivalence relation and give the equivalence classes. There is some practice in the textbook for this.
Notice in the example we have given, we get five sets as the partition. One of these sets is a subgroup of \(\Z\) (which one). The others are not. But the others can be described in terms of this subgroup. This is what we are really after here.
In fact, could we describe the structure of these five sets using tools we have already developed? Might these five sets interact with each other in a group or ring sort of way? What might this even mean?
Subsection Wednesday 10/22
Today we will explore the concept of cosets, the partition of a group induced by a subgroup, and under what conditions the set of cosets form a group themselves.
Work through the Cosets are my bags activity.
Handout Friday 10/24
Today we will explore the structure of cosets and what that tells us about subgroups.
Let’s do an example. Take a subgroup \(H= \{(1), (1234), (13)(24), (1432)\}\) of \(S_4\text{.}\) What are the cosets of the subgroup?
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Write each coset as a left coset: \(aH\) for \(a \in S_4\text{.}\) For example, \((123)H\text{.}\) What is this? Note that this is NOT the same set as the corresponding right coset.
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What do we notice? How many elements are in this coset?
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One of the elements in this coset is \((1342)\text{.}\) What elements are in the coset \((1342)H\text{?}\)
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In fact, if \(b \in aH\) the \(aH = bH\text{.}\) The converse is also true (this should be obvious?)
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What can we conclude from these two facts? The cosets will form a partition of the group.
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But also, every coset will have the same size! That means that...
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The size of \(H\) must divide the size of the group.
Theorem 2. Lagrange.
Let \(G\) be a finite group and let \(H\) be a subgroup of \(G\text{.}\) Then \(|G|/|H| = [G : H]\) is the number of distinct left cosets of \(H\) in \(G\text{.}\) In particular, the number of elements in \(H\) must divide the number of elements in \(G\text{.}\)
Proof.
The group \(G\) is partitioned into \([G : H]\) distinct left cosets. Each left coset has \(|H|\) elements; therefore, \(|G| = [G : H] |H|\text{.}\)
A cool consequence of this theorem is that it also tells us something about elements. The order of an element \(a\) is the least \(k\) such that \(a^k = e\) (the identity). Since \(\langle a \rangle\) is a subgroup, the order of the subgroup must be a divisor of the size of the group. But the order of the cyclic subgroup is the same as the order of its generator. So orders of elements must also divide the size of the group.
In particular, suppose a group \(G\) has an element \(a\) such that \(a^7 = e\text{.}\) Now this means that either \(a = e\) or the order of \(a\) is 7. That means that \(G\) must have size that is a multiple of 7.
