What does it mean for an element \(g\) to generate a group \(G\)?

In multiplicative notation, what is \(g^0\)?

If \(r\) is rotation by \(90^\circ\), what rotation is \(r^2\)?

If \(g^5=e\) is the first positive return to the identity, what is the order of \(g\)?

List the distinct powers before repetition if \(g^4=e\) first occurs at exponent \(4\).

For rotations of an equilateral triangle generated by \(r\), where \(r\) is \(120^\circ\), list \(\langle r\rangle\).

In a cyclic group with \(g^6=e\), reduce \(g^8\) to one of \(e,g,g^2,g^3,g^4,g^5\).

In \(\mathbb Z_5\) under addition, what is generated by \(1\)?

In \(\mathbb Z_6\) under addition, list the elements generated by \(2\).

For a regular pentagon, let \(r\) be rotation by \(72^\circ\). What is the order of \(r\)?

Show that every element of a cyclic group commutes with every other element.

In a cyclic group generated by \(g\) with \(g^7=e\), find the inverse of \(g^3\).

Determine whether \(3\) generates \(\mathbb Z_8\) under addition.

Determine whether \(4\) generates \(\mathbb Z_{10}\) under addition.

If \(g\) has order \(12\), find the order of \(g^4\).

If \(g\) has order \(12\), find the order of \(g^5\).

Explain why stopping after seeing several distinct powers is not enough to prove an element generates a finite group.

Prove that if \(g\) has order \(n\), then \(g^{-1}=g^{n-1}\).

A student says a group of order \(6\) must be cyclic because repeatedly applying any non-identity element eventually repeats. Diagnose the error.

Show that every subgroup of a cyclic group is generated by powers of the same generator in the finite case, using the smallest positive exponent idea.