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Q)

# Show that the set {, , , , } forms an abelian group under multiplication modulo 11. Comment
A)
Toolbox:
• Multiplication modulo n $(._{\large n}):a._{\large n} b=r ; 0\leq r< n$,where r is the least non-negative remainder when $ab$ is divided by n.
• Congruence modulo n :Let $a, b \in Z$ and $n$ be a fixed positive integer.
• We say that “a is congruent to b modulo n” $\Leftrightarrow (a − b)$ is divisible by n Symbolically,
• $a \equiv b (mod\; n) \Leftrightarrow (a − b)$ is divisible by n.
Step 1:
Let $G=\{,,,,\}$
Consider $(G,._{11})$
Drawing up the multiplication table,we have
Step 2:
Closure :All the products in the table belong to $G$.Closure is satisfied.
Associativity : Modular multiplication is associative.Since multiplication of integers is associative.
Existence of identity :From the table,it is evident that  is the identity element.
Existence of inverse: From the table it can be seen that the inverse of  is ,$^{-1}$ is ,$^{-1}$ is ,$^{-1}$ is  and $^{-1}$ is .All the elements of $G$ have inverse in G.
The four group axioms being satisfied.,$(G,._{11})$ is a group.
Step 3:
Commutative property : Modular multiplication is commutative since multiplication of integers is commutative.
$\therefore (G,._{11})$ is an abelian group.