The following is a multiplication table for a group ( G ). The product ( a * b ) is The following is a multiplication table for a group G. The product a**b is given in the
table where a is from the leftmost column and b from the top row.
(a) What is the order of the group?
(b) Is the group Abelian? Give a reason for your answer.
(c) What are the inverses of the elements?
(d) What are the orders of the elements?
(e) Let H={e,z}. Show that this is a subgroup of G.
(f) Calculate the right cosets of H.
(g) Without calculating the left cosets of H, explain why H is a normal subgroup of
G.
(h) Construct a multiplication table for the quotient group (G)/(H).
(i) Is (G)/(H) cyclic? Give reasons.
(j) Describe the subgroup Z<=(G)/(H) generated by the element Hxin(G)/(H).
(k) What is the order of ((G)/(H))/(Z) ?
(a) Prove that if {v_(1),v_(2),v_(3)} is a linearly independent subset of a real vector space,given in the table where ( a ) is from the leftmost column and ( b ) from the top row. egin{tabular}{c|cccccc} & ( e ) & ( v ) & ( w ) & ( x ) & ( y ) & ( z ) \ hline( e ) & ( e ) & ( v ) & ( w ) & ( x ) & ( y ) & ( z ) \ ( v ) & ( v ) & ( y ) & ( e ) & ( w ) & ( z ) & ( x ) \ ( w ) & ( w ) & ( e ) & ( x ) & ( z ) & ( v ) & ( y ) \ ( x ) & ( x ) & ( w ) & ( z ) & ( y ) & ( e ) & ( v ) \ ( y ) & ( y ) & ( z ) & ( v ) & ( e ) & ( x ) & ( w ) \ ( z ) & ( z ) & ( x ) & ( y ) & ( v ) & ( w ) & ( e ) end{tabular} (a) What is the order of the group? (b) Is the group Abelian? Give a reason for your answer. (c) What are the inverses of the elements? (d) What are the orders of the elements? (e) Let ( H={e, z} ). Show that this is a subgroup of ( G ). (f) Calculate the right cosets of ( H ). (g) Without calculating the left cosets of ( H ), explain why ( H ) is a normal subgroup of ( G ). (h) Construct a multiplication table for the quotient group ( G / H ). (i) Is ( G / H ) cyclic? Give reasons. (j) Describe the subgroup ( Z leq G / H ) generated by the element ( H x in G / H ). (k) What is the order of ( (G / H) / Z ) ? (a) Prove that if ( left{mathbf{v}_{1}, mathbf{v}_{2}, mathbf{v}_{3}
ight} ) is a linearly independent subset of a real vector space,
The following is a multiplication table for a group G. The product a * b is given in the table where a is from the leftmost column and b from the top row.
e
v W x y e e w x y 2 v y e w x w w e x v y x x w y e y y e x m x y w e
(a) What is the order of the group?
(b) Is the group Abelian? Give a reason for your answer. (c) What are the inverses of the elements? (d) What are the orders of the elements? (e) Let H = {e, z}. Show that this is a subgroup of G.
(f) Calculate the right cosets of H. (g) Without calculating the left cosets of H, explain why H is a normal subgroup of G.
(h) Construct a multiplication table for the quotient group G/H
(i) Is G/H cyclic? Give reasons. (j) Describe the subgroup Z < G/H generated by the element Hx E G/H. (k) What is the order of (G/H)/Z?
(a) Prove that if {v1,V2,V} is a linearly independent subset of a real vector space then so is v.
