Subgroups and cyclic groups

Example 1.2. (i) For every group G, G ≤ G. If H ≤ G and H 6= G, we call H a proper subgroup of G. Similarly, for every group G, {1} ≤ G. We call {1} the trivial subgroup of G. Most of the time, we are interested in proper, nontrivial subgroups of a group. (ii) Z ≤ Q ≤ R ≤ C; here the operation is necessarily addition. Similarly, Q∗ ≤ R∗ ≤ C∗, where the operation is multiplication. Likewise, μn ≤ U(1), for every n, and U(1) ≤ C∗.