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Unformatted text preview: 98 2. BASIC THEORY OF GROUPS 2Œ4 D Œ8, 3Œ4 D Œ12, 4Œ4 D Œ2, 5Œ4 D Œ6, 6Œ4 D Œ10, 7Œ4 D Œ0. So
the order of Œ4 is 7.
Example 2.2.19. What is the order of Œ5 in ˚.14/? We have Œ52 D Œ11,
Œ53 D Œ13, Œ54 D Œ9, Œ55 D Œ3, Œ56 D Œ1, so the order of Œ5 is 6.
Note that j˚.14/j D '.14/ D 6, so this computation shows that ˚.14/ is
cyclic, with generator Œ5.
The next result says that cyclic groups are completely classiﬁed by
their order. That is, any two cyclic groups of the same order are isomorphic.
Proposition 2.2.20. Let a be an element of a group G .
(a) If a has inﬁnite order, then hai is isomorphic to Z.
(b) If a has ﬁnite order n, then hai is isomorphic to the group Zn .
Proof. For part (a), deﬁne a map ' W Z ! hai by '.k/ D ak . This
map is surjective by deﬁnition of hai, and it is injective because all powers
of a are distinct. Furthermore, '.k C l/ D ak Cl D ak al . So ' is an
isomorphism between Z and hai.
For part (b), since Zn has n elements Œ0; Œ1; Œ2; : : : ; Œn 1 and hai has
n elements e; a; a2 ; : : : ; an 1 , we can deﬁne a bijection ' W Zn ! hai by
'.Œk/ D ak for 0 Ä k Ä n 1. The product (addition) in Zn is given by
Œk C Œl D Œr, where r is the remainder after division of k C l by n. The
multiplication in hai is given by the analogous rule: ak al D ak Cl D ar ,
where r is the remainder after division of k C l by n. Therefore, ' is an
isomorphism.
I Subgroups of Cyclic Groups
In this subsection, we determine all subgroups of cyclic groups. Since
every cyclic group is isomorphic either to Z or to Zn for some n, it sufﬁces
to determine the subgroups of Z and of Zn .
Proposition 2.2.21.
(a) Let H be a subgroup of Z. Then either H D f0g, or there is a
unique d 2 N such that H D hd i D d Z.
(b) If d 2 N , then d Z Š Z.
(c) If a; b 2 N , then aZ Â b Z if, and only if, b divides a. Proof. Let’s check part (c) ﬁrst. If aZ Â b Z, then a 2 b Z, so b divides
a. On the other hand, if b ja, then a 2 b Z, so aZ Â b Z. ...
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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
 EVERAGE
 Algebra

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