Unformatted text preview: 101 2.2. SUBGROUPS AND CYCLIC GROUPS Z12 d
d hŒ3i hŒ2i d d
d d hŒ6i hŒ4i d
d f0g Figure 2.2.2. Lattice of subgroups of Z12 . integers s such that s is congruent modulo n to an integer multiple of b ,
or, equivalently, the smallest of positive integers s such that Œs 2 hŒbi.
By the proof of Proposition 2.2.24, hŒd i D hŒbi. The order of Œb is the
order of hŒbi, which is n=d , by Proposition 2.2.24 (b). Part (c) is left as
Example 2.2.29. Find all generators of Z12 . Find all Œb 2 Z12 such that
hŒbi D hŒ3i, the unique subgroup of order 4. The generators of Z12 are
exactly those Œa such that 1 Ä a Ä 11 and a is relatively prime to 12.
Thus the generators are Œ1; Œ5; Œ7; Œ11. The generators of hŒ3i are those
elements Œb satisfying g:c:d:.b; 12/ D g:c:d:.3; 12/ D 3; the complete list
of these is Œ3; Œ9.
The results of this section carry over to arbitrary cyclic groups, since
each cyclic group is isomorphic to Z or to Zn for some n. For example,
Proposition 2.2.30. Every subgroup of a cyclic group is cyclic. Proposition 2.2.31. Let a be an element of ﬁnite order n in a group. Then
hak i D hai, if, and only if, k is relatively prime to n. The number of
generators of hai is '.n/. ...
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