1.Let G= <{a,b,c,d},*> and the operations are shown in thetable below:
a
b
c
d
a
a
b
c
d
b
b
c
d
a
c
c
d
a
b
d
d
a
b
c
Show that, G is an abelian group.
Answer
For a group G to be an abelian group, t must satisfy therequirements-
1.CLOSURE i.e, the result of the operationshould belong to G.
In the given table we can see that theresult is {a,b,c,d} which is the input itself, therefore itsatisfies closure property.
2.ASSOCIATIVITY which means that x*(y*z) =(x*y)*z
Let’s say for an example we take x=a,y=b and z=c.
on LHS we will have, a*(b*c) = a*d =d
on RHS we will have (a*b)*c = b*c =d
This is applicable for all valuesbelonging to
OR
OR