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Introduction to Abstract Algebra I MATH 321 Fall 2025

Worksheet Homework 3

1.

Consider the permutation \(\alpha = (1435)(267)\) in the group \(S_7\text{.}\)
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(a)

Find \(\alpha\inv\) and verify your answer is correct. Clearly explain how you perform the computation.
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(b)

Compute \(\alpha (123) \alpha\inv\text{.}\)
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(c)

Find \(\alpha^2\text{,}\) and \(\alpha^3\text{.}\) Will there be some \(n\) such that \(\alpha^n = (1)\text{?}\) Explain why not or find the least such \(n\text{.}\)
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2.

Let \(G\) be a group, writing the operation for \(G\) as multiplication and \(e\) the identity. Which of the following must be true. If the statement is true, prove it. If it is not always true, find a specific example of a group and specific elements in that group that give a counterexample.
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(a)

For all \(a, b, c \in G\text{,}\) if \(aba\inv = c\) then \(b = c\)
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(b)

For all \(a, b \in G\text{,}\) if \(aba\inv = e\) then \(b = e\text{.}\)
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(c)

For any elements \(a, b \in G\text{,}\) if \(ab^2 = e\) then \(a = e\) or \(b = e\text{.}\)
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(d)

For any elements \(a, b \in G\text{,}\) if \(ab^2 = e\) then \(a\) and \(b\) commute.
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3.

We claimed that all group tables satisfy the “sudoku rule,” by which we mean that in every row and every column each element of the group must appear exactly once. Prove this!
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Hint 1.
There are two things to prove: first that every element must appear in each row and each column, and second, that no element appears more than once in any row or column.
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Hint 2.
Think about what equation you would write down to say that \(a\) appears in \(b\)’s row. There must be some element \(c\) such that...
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4.

In which sorts of rings \(R\) is it true that for any \(a \in R\text{,}\) if \(a^3 = 0\) then \(a = 0\text{?}\) Prove your answer (including giving an example of a ring that is not of this sort in which the statement is not true.)
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