[數學]question on Groups --PLEASE HELP ME

[數學]question on Groups --PLEASE HELP ME

jocelyn 於 星期三 十月 11, 2006 10:09 am


1. a) Let x,y ∈G be two elements of a group G, and put z =x -1yx. Show that
        zn =x -1ynx ,for all n∈integers and conclude that ord(z)=ord(y).
    b) Let c =( i1, i2 , i3 ,......,ir)∈Sn be an r-cycle. If α ∈Sn ,Show that
        c' = αcα-1 is also an r-cycle by verifying that c'= (α (i1 ) α (i2 ).......α(ir) )  
 
2. Let G be a group, and put Z(G) ={x ∈G : xy=yx, for all y∈G}
    a) Show that Z(G) is an abelian subgroup of G.
    b) Verify that Z(G)=G if and only if G is abelian.
 
3. (a) List all the elements of the alternating group A4 ,and find all its cyclic subgroups.
    (b) Let α=(12)(34) and β =(123), and put H=(α,β). Show that V ≤H and conclude
         (using Lagrange's Theorem) that H=G.

jocelyn
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