Question

Find the elementary divisors and the invariant factors of the given group. Note that the group operation is multiplication in the first three and addition in the last.

(a) (b) (c) (d)

Short answer

Invariant Factors:

Ifis a finitely generated abelian group, then

,

wherefor some integerwe have,

Such thatand

Thus, the integersare known as theinvariant factors of the group.

Elementary Divisors:

Ifis in the direct sum of cyclic groups of prime orders the prime powers are known as elementary divisorsof.

Step-by-step solution

Write the definition of invariant factors of  and elementary divisors of   

Invariant Factors:

Ifis a finitely generated abelian group, then

,

wherefor some integerwe have,

Such thatand

Thus, the integersare known as theinvariant factors of the group.

Elementary Divisors:

Ifis in the direct sum of cyclic groups of prime orders the prime powers are known as elementary divisorsof.

Solution of part (a)

Consider.

As we know we have

Thus, the requiredelementary divisors areand invariant factorsare.

Solution of part (b)

Consider.

As we know we have

Thus, the required elementary divisors areand invariant factorsare.

 Solution of part (c) 

Consider.

As we know we have

Thus, the required elementary divisors areand invariant factorsare.

Solution of part (d) 

Consider.

As we know,we have,

Thus, the required elementary divisors areand invariant factors are:.