Exercise 478Algebra › GroupsHigh schoolAdded 3 Nov 2026

Which subsets are subgroups?

#subgroup#examples#counterexamples

Statement

Determine whether each subset is a subgroup:

(a) $2\mathbb{Z}\subseteq(\mathbb{Z},+)$ ;
(b) $\mathbb{N}\subseteq(\mathbb{Z},+)$ ;
(c) $\{1,-1\}\subseteq(\mathbb{R}^*,\times)$ ;
(d) $\{z\in\mathbb{C}:|z|=1\}\subseteq(\mathbb{C}^*,\times)$ .

Hint

Check the subgroup criterion: nonempty, closed under product and inverse.

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Solution

(a) $2\mathbb{Z}$ : $0\in2\mathbb{Z}$ ; sum of evens is even; inverse of even is even. Subgroup.

(b) $\mathbb{N}$ : closed under sum, but no inverse for positive integers. Not a subgroup.

(c) $\{1,-1\}$ : products stay in the set, inverses are themselves. Subgroup.

(d) $|z|=|w|=1\Rightarrow|zw|=1$ and $|z^{-1}|=1$ . Subgroup (the circle group).

Conclusion. (a), (c), (d) are subgroups; (b) is not.

Which subsets are subgroups?

Statement

Solution

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