Introduction to Abstract Algebra I MATH 321 Fall 2025

Let \(G\) be a group with subgroup \(H\text{.}\) Define the relation \(\sim\) on \(G\) by \(x \sim y\) if and only if \(y\inv x \in H\text{.}\) Prove that \(\sim\) is an equivalence relation.

In \(S_4\text{,}\) let \(H = \{(1), (1234), (13)(24), (1432), (12)(34), (14)(23), (13), (24)\}\text{.}\) (This subgroup is really \(D_4\text{,}\) by the way.) Illustrate the result in exercise 1 with this example. That is, give a few examples of elements that are equivalent and some that are not. Then find the three equivalence classes for \(\sim\text{,}\) i.e., find three different \([a] = \{b \in S_4 \st b \sim a\}\) for three choices of \(a \in S_4\text{.}\)

Finally, how do these sets relate to the cosets of \(H\) in \(S_4\text{?}\)

In exercise 1, we require that \(H\) is a subgroup of \(G\text{.}\) What goes wrong if it is not? Work in \(G = \Z_{7}\) (under addition) and let \(H = \{2, 4, 6\}\text{.}\) Show explicitly what goes wrong if you wanted \(\sim\) to be an equivalence relation (which of the three properties of an equivalence relation are satisfied). Then show explicitly the way that the equivalence classes fail to form a partition.

Finally, what happens if you try to form cosets \(a + H\) for \(a \in \Z_7\text{.}\)

Let \(G\) be a group and \(H\) a subgroup of \(G\text{.}\) Prove for any \(a, b \in G\text{,}\) that \(aH = bH\) if and only if \(b\inv aH = H\text{.}\)

It appears like we can operate with cosets the same as we can with group elements, i.e., we can treat \(H\) like an element of \(G\) instead of a set of elements of \(G\text{.}\) This is mostly true. Let’s explore a couple of examples.

Suppose \(H\) is a subgroup of \(G\) and \(a, b, c \in G\text{.}\) Prove that \(abH = acH\) if and only if \(bH = cH\text{.}\)

Then prove, by finding a counterexample, that it is possible for \(aH = bH\) even though \(a \ne b\) (that is, you can’t cancel the \(H\)).