Understanding Cyclic Groups and Theorems in Group Theory
Explore the concept of cyclic groups, generator elements, theorems in group theory, and the group of all integers modulo n. Learn about abelian groups, congruence modulo n, and exercises related to commutative groups.
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Presentation Transcript
Cyclic group Def: let(G,*) be gp & a G , if G =(a) where (a)={??;k Z} then G is called cyclic gp generated by a & a is called generator element
example 1- in (Z,+) ,{-1 , 1} are generator element 2- in( {1,-1,I,-i}.),{i, -i} are generator element So (Z,+) & {1,-1,I,-i}are cyclic group
theoremes in any group (G,*) 1- (a) =(? 1) 2- (e )= { e} 3- if (G,*) is cyclic then (G,*) is abelian But the converes of 3 is not true for example: ({e,a,b,c},.);?2=?2=?2=e is abelian group but not cyclic
The group of All integers module of n;n Z+ Def: let n Z+ & a,b Z,a is said to be congurent to b modul of n iff a-b=kn ,for some k Z and denoted by a b (mod n) . example: 47 (-1) (mod 12) since 47-(-1)=48
exercies 1-(Zn , n) is commutative gp 2- is (Zn, n) gp 3- is (Z6, 6) gp 4- (Zn\[0], n) is gp iff n is aprime no.
