MA8551 / ANT - MODEL

When the group G is called Abelian group

• It's satisfies Closure

• It's satisfies Commutative

• It's satisfies Associative

For every group G, if  $a,b,c\in G$  and  $ab=ac$  then  $b=c$  

• Left cancellation law

• Right cancellation law

• Abelian group

When the set G is written as $\left(ab\right)^{-1}=a^{-1}b^{-1}$ for all  $a,b\in G$  

• G is a group

• G is an abelian

• G is commutative

Define Subgroup

• A empty subset H of a group G is called a sub-group of G whenever H itself a group under the binary operation in G

• A non-empty subset H of a group G is called a sub-group of G whenever H itself a group under the binary operation in G

• A non-empty subset H of a group G is called a sub-group of G

When $H\cap K$  is called Subgroup of G

• H is Subgroup of G

• K is subgroup of G

• Both H and K are subgroup of G

Which one is the subgroups of $\left(Z_{11}^*,\ \circ\right)$  

• {1, 10}

• {1, 3, 4, 5, 9}

• both

Which one is the subgroups of $\left(Z_{11}^*,\ \circ\right)$  

• {1, 10}

• {1, 3, 4, 5, 9}

• both

 $f_1$  is the identity we can rotate  $90\degree$  we get  $f_2$  . what is the order of  $f_2$  

• 4

• 2

• 1

Which one is the properties of group homomorphism

• 
  

  $ab=ba$
  

• $f\left(a^n\right)=\left[f\left(a\right)\right]^n$

• $a+b=ab$

Every subgroup of a cyclic group is

• group

• subgroup

• cyclic