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Unformatted text preview: 97 2.2. SUBGROUPS AND CYCLIC GROUPS S3
¨¨ ¨¨ ¨ ¨¨ ¨ ¨¨ h.12/i rr h.13/i d
d h.23/i d
d h.123/i fe g Figure 2.2.1. Lattice of subgroups of S3 . Example 2.2.15. The set of all powers of r in the symmetries of the square
is fe; r; r 2 ; r 3 g.
There are two possibilities for a cyclic group hai, as we are reminded
by these examples. One possibility is that all the powers ak are distinct, in
which case, of course, the subgroup hai is inﬁnite; if this is so, we say that
a has inﬁnite order.
The other possibility is that two powers of a coincide. Suppose k < l
and ak D al . Then e D .ak / 1 al D al k , so some positive power of a
is the identity. Let n be the least positive integer such that an D e . Then
e; a; a2 ; : : : ; an 1 are all distinct (Exercise 2.2.9) and an D e . Now any
integer k (positive or negative) can be written as k D mn C r , where the
remainder r satisﬁes 0 Ä r Ä n 1. Hence ak D amnCr D amn ar D
e m ar D ear D ar . Thus hai D fe; a; a2 ; : : : ; an 1 g. Furthermore,
ak D al if, and only if, k and l have the same remainder upon division by
n, if, and only if, k Á l .mod n/.
Deﬁnition 2.2.16. The order of the cyclic subgroup generated by a is
called the order of a. We denote the order of a by o.a/.
The discussion just before the deﬁnition establishes the following assertion:
Proposition 2.2.17. If the order of a is ﬁnite, then it is the least positive
integer n such that an D e . Furthermore, hai D fak W 0 Ä k < o.a/g.
Example 2.2.18. What is the order of Œ4 in Z14 ? Since the operation in
the group Z14 is addition, powers of an element are multiples. We have ...
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This note was uploaded on 12/15/2011 for the course MAC 1105 taught by Professor Everage during the Fall '08 term at FSU.
- Fall '08