GROUP THEORY & PROBABILITY THEORY

If A and B are two elements of a group G under multiplication then the element of a G must be

• $A+B$

• $A-B$

• $AB$

• $\frac{A}{B}$

If any two elements A and B of a group commute then the group is

• Cyclic group

• $abelian$

• Non abelian

• Permutation group

If E is identity element of group G and A any element in the same group G then

• $A+E=A$

• $AE=0$

• $AE=A$

• $AE=E$

If all elements of a group may be expressed by the powers of a single element the group is called

• Cyclic

• Subgroup

• Non Abelian

• Power Group

Two groups  $G=(G_1,G_2,……….G_n)$  and  $H=(H_1,H_2,.......,H_n)$  are isomorphic if 

•  $G_1H_1=G_2H_2$  

•  $\frac{G_1}{H_1}=\frac{G_2}{H_2}$  

•  $G_1G_2=H_1H_2$  

•  $\frac{G_1}{G_2}=\frac{H_1}{H_2}$  

Two groups are isomorphic if

• same number of elements and same multiplication table

• same multiplication table different number of elements

• same number of elements and different multiplication table

• different number of elements and different multiplication table

Three cube roots of unity form

• An abelian under addition

• An abelian under multiplication

• Non abelian under addition

• Non abelian under multiplication

The factor group of an abelian is necessarily

• An abelian

• Cyclic

• Non abelian

• Permutation

The group of order 3 is

• Always a cyclic

• Never a cyclic

• May or may not be a cyclic

• Does not contain identity element

The group of order 4 is

• always a cyclic

• never a cyclic

• may or may not be a cyclic

• does not contain identity element