YLL討論網

#ed_op#DIV#ed_cl#1. a) Let x,y ∈G be two elements of a group G, and put z =x #ed_op#SUP#ed_cl#-1#ed_op#/SUP#ed_cl#yx. Show that#ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl#        z#ed_op#SUP#ed_cl#n #ed_op#/SUP#ed_cl#=x #ed_op#SUP#ed_cl#-1#ed_op#/SUP#ed_cl#y#ed_op#SUP#ed_cl#n#ed_op#/SUP#ed_cl#x ,for all n∈integers and conclude that ord(z)=ord(y).#ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl#    b) Let c =( i#ed_op#SUB#ed_cl#1#ed_op#/SUB#ed_cl#, i#ed_op#SUB#ed_cl#2 #ed_op#/SUB#ed_cl#, i#ed_op#SUB#ed_cl#3 #ed_op#/SUB#ed_cl#,......,i#ed_op#SUB#ed_cl#r#ed_op#/SUB#ed_cl#)∈Sn be an r-cycle. If α ∈Sn ,Show that#ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl#        c' = αcα#ed_op#SUP#ed_cl#-1#ed_op#/SUP#ed_cl# is also an r-cycle by verifying that c'= (α (i#ed_op#SUB#ed_cl#1#ed_op#/SUB#ed_cl# ) α (i#ed_op#SUB#ed_cl#2#ed_op#/SUB#ed_cl# ).......α(i#ed_op#SUB#ed_cl#r#ed_op#/SUB#ed_cl#) )  #ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl# #ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl#2. Let G be a group, and put Z(G) ={x ∈G : xy=yx, for all y∈G} #ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl#    a) Show that Z(G) is an abelian subgroup of G.#ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl#    b) Verify that Z(G)=G if and only if G is abelian.#ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl# #ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl#3. (a) List all the elements of the alternating group A#ed_op#SUB#ed_cl#4 #ed_op#/SUB#ed_cl#,and find all its cyclic subgroups.#ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl#    (b) Let α=(12)(34) and β =(123), and put H=(α,β). Show that V ≤H and conclude #ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl#         (using Lagrange's Theorem) that H=G. #ed_op#/DIV#ed_cl#