In a group table, what do the row label \(a\) and column label \(b\) represent?

What feature of a group table shows closure?

For a table with headings \(e,a\), the row \(e\) reads \(e,a\). What does this show about left multiplication by \(e\)?

If the entry in row \(a\), column \(b\) is \(e\), what inverse information does this provide?

Use the table row \(a\): entries under columns \(e,a\) are \(a,e\). What is \(a*a\)?

For \(G=\{e,a\}\) with \(a*a=e\), find \(a^{-1}\).

A table has headings \(e,a,b\). The row \(e\) is \(e,a,b\), but the column \(e\) is \(e,b,a\). Can \(e\) be a two-sided identity?

In a table for \(G=\{e,a,b\}\), row \(a\), column \(b\) is \(e\), and row \(b\), column \(a\) is \(e\). What can you conclude?

For addition modulo \(3\), compute the row for element \(1\) with columns \(0,1,2\).

For addition modulo \(4\), find the inverse of \(3\) by table reasoning.

Why must each row of a finite group table contain the identity exactly once?

A table has a row with a repeated entry. Explain why it cannot be a group table.

Build the Cayley table for \(\{0,1,2\}\) under addition modulo \(3\). Give the rows in order \(0,1,2\).

Given a finite group table, describe how to find all self-inverse elements.

A table is symmetric across its main diagonal. What does that suggest about commutativity, and what does it not prove?

A finite table has every row and column containing each element exactly once. Why is this still not a complete proof that the table defines a group?

Explain how row-first convention affects reading a non-commutative group table.

A student identifies an identity by finding only a row that reproduces the headings. Diagnose the mistake.

Prove that each column of a finite group table contains every group element exactly once.

A proposed table for four elements has one entry outside the listed set but otherwise looks like a group table. What is the decisive failure?