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Introduction to Abstract Algebra I MATH 321 Fall 2025

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Handout Wednesday 10/15

Today we will wrap up our look into cyclic subgroups and then start thinking about what it means for a subgroup to partition a group.
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What are the cyclic subgroups of \(S_5\text{?}\) Is any one of them \(S_5\) itself? That is, will \(S_5\) be cyclic?
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Now let’s go back to our subgroups of \(\Z_{12}\text{.}\) In particular, look at \(H = \{0, 4, 8\} = \langle 4\rangle\text{.}\) We would like to create a partition of \(\Z_{12}\) that is somehow based on \(H\text{.}\)
  • What does “partition” even mean? We mean a collection of sets (we might call these blocks) such that
    1. Every element is in one of the sets (blocks).
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    2. No element is in more than one set (block).
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    For example, a partition of \(\Z_{12}\) could be
    \begin{equation*} \{\{0,4,8\}, \{1,2,3\}, \{5, 6, 7, 9, 10\}, \{11\}\}\text{.} \end{equation*}
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  • A nicer partition might be
    \begin{equation*} \{\{0, 4, 8\}, \{1, 5, 9\}, \{2, 6, 10\}, \{3, 7, 11\}\}\text{.} \end{equation*}
    (How did I come up with this?)
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  • Before we get too deep into examples involving groups, we should think about partitions more generally.
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  • Another example: Let \(A_k = \{n \in \Z \st n \equiv k \pmod{5}\}\text{.}\) Is \(\{A_k\}_{k \in \Z}\) a partition of \(\Z\text{?}\) Note that \(A_3 = A_8\text{.}\) In fact, how many sets are in the partition?
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So a partition is a way to divide up the elements in a set. This is closely related to the idea of an equivalence relation. What is this?
  • The idea is that we want to axiomatize “equals”. What are the fundamental properties of the equal relation? The relation satisfies three properties:
    1. Reflexive: For all \(x\text{,}\) \(x = x\text{.}\)
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    2. Symmetric: For all \(x\) and \(y\text{,}\) if \(x = y\) then \(y = x\text{.}\)
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    3. Transitive: For all \(x\text{,}\) \(y\text{,}\) and \(z\text{,}\) if \(x = y\) and \(y = z\text{,}\) then \(x = z\text{.}\)
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  • Other relations satisfy these three properties, and when they do, we say that the relation is an equivalence relation.
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  • For example, congruence mod 5 is an equivalence relation (on the integers).
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  • How are equivalence relations related to partitions? Well any equivalence relation gives rise to a partition by considering the equivalence classes of an element. That is, we define \([a] = \{b \st a \equiv b\}\text{,}\) the set of all elements that are equivalent to \(a\text{,}\) and call this the equivalence class of \(a\).
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  • The equivalence classes form a partition.
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  • Conversely, given any partition, we can define an equivalence relation by saying two elements are equivalent if they belong to the same set in the partition.
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  • These two operations are inverses of each other. If you start with a partition, define the equivalence relation, and then take the equivalence classes for that equivalence relation, you get the original partition back. And similarly if you start with the equivalence relation.
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