Exercise · ZW-SUBGROUP-010 · ★★★ Advanced
Let \(H\) be a finite, nonempty subset of a group \(G\), closed under the operation. Is \(H\) a subgroup ?
Choose one option.
A Yes: the powers of each \(a\in H\) repeat, and from that \(a^{-1}\in H\) B No: one must explicitly require inverses C Only if \(G\) is abelian D Only if \(|H|\) is prime Show answer A. Yes: the powers of each \(a\in H\) repeat, and from that \(a^{-1}\in H\)← Back to the question
Explanation Since \(H\) is finite, the powers \(a,a^2,\dots\) repeat: \(a^i=a^j\) with \(i<j\), so \(a^{j-i}=e\) and \(a^{-1}=a^{j-i-1}\in H\). For infinite sets this fails, as in \(\mathbb{N}\subset\mathbb{Z}\).
Review the concept: Subgroup A subset \(H\) of a group \(G\) that is itself a group under the operation of \(G\).
\[H\le G\]
More practice problems: subgroup