# Generators of Z6

by ma3088
Tags: abstarct algebra, generators
 Sci Advisor HW Helper PF Gold P: 3,216 Your notation is incorrect. {1} is a set containing one element, 1. {1, 2, 3, 4, 5, 0} is a set containing six elements. Therefore {1} does not equal {1, 2, 3, 4, 5, 0}. The usual notation for the group generated by a set is a pair of angle brackets: <{1}> denotes the group generated by the set {1}. It is true that <{1}> = <{5}> = Z6. It is also (trivially) true that <{1, 2, 3, 4, 5, 0}> = Z6. Note that in general, if S is a subset of Z6, is the smallest subgroup of Z6 which contains all of the elements of S. If S is a subgroup, then S = . Also, it's easy to verify that if S $\subseteq$ T, then $\subseteq$ . Now what about <{2,3}>? This is a group, by definition, so it must be closed under addition. Thus <{2,3}> must contain 5 because 2+3=5. In other words, {5} $\subseteq$ <{2,3}>. Therefore Z6 = <{5}> $\subseteq$ <{2,3}>. For the reverse containment, we have {2,3} $\subseteq$ {1,2,3,4,5,0}, so <{2,3}> $\subseteq$ <{1,2,3,4,5,0}> = Z6. We conclude that <{2,3}> = Z6.