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.

1. U₁₈ (b) U₁₇ (c) U₁₅ (d)$\mathbf{M}\left({\mathbb{Z}}_2\right)$

Short answer

1. The elementary divisors are {2,2} and invariant factors are {2,2}.
2. The elementary divisors are {1,6} and invariant factors are {2,4}.
3. The elementary divisors are {2,4} and invariant factors are {2,4}.
4. The elementary divisors are {2,2,2,2} and invariant factors are {2,2,2,2}.

Step-by-step solution

Write the definition of invariant factors of G and elementary divisors of G

Invariant Factors:

If G is a finitely generated abelian group, then

$\mathrm{G}\cong {\mathrm{\mathbb{Z}}}^{\mathrm{r}}\oplus {\mathrm{\mathbb{Z}}}_{{\mathrm{m}}_1}\oplus ....\oplus {\mathrm{\mathbb{Z}}}_{{\mathrm{m}}_{\mathrm{s}}}$,

where for some integer r we have,${\mathrm{m}}_1\ge {\mathrm{m}}_2\ge .....\ge {\mathrm{m}}_{\mathrm{s}}\ge 2$

Such that$r\ge 0$ and$\frac{m_{i+1}}{m_i}$

Thus, the integers${\mathrm{m}}_1,{\mathrm{m}}_2,....,{\mathrm{m}}_{\mathrm{s}}$ are known as the invariant factors of the group.

Elementary Divisors:

If G is in the direct sum of cyclic groups of prime orders the prime powers are known as elementary divisors of G.

Solution of part (a)

Consider ${\mathrm{U}}_8$.

As we know we have$U_8\cong {\mathrm{\mathbb{Z}}}_2\oplus {\mathrm{\mathbb{Z}}}_2$

Thus, the required elementary divisors are {2,2} and invariant factors are {2,2}.

Solution of part (b)

Consider $U_{17}$.

As we know we have$U_{17}\cong {\mathrm{\mathbb{Z}}}_{2^4}$

Thus, the required elementary divisors are {16} and invariant factors are {2,4}.

Solution of part (c)

Consider U₁₅.

As we know we have${\mathrm{U}}_{15}\cong {\mathrm{\mathbb{Z}}}_2\oplus {\mathrm{\mathbb{Z}}}_{2^2}$

Thus, the required elementary divisors are {2,4} and invariant factors are {2,4}.

Solution of part (d)

Consider $\mathrm{M}\left({\mathrm{\mathbb{Z}}}_2\right)$.

As we know, we have,$\mathrm{M}\left({\mathrm{\mathbb{Z}}}_2\right)\cong {\mathrm{\mathbb{Z}}}_2\oplus {\mathrm{\mathbb{Z}}}_2\oplus {\mathrm{\mathbb{Z}}}_2\oplus {\mathrm{\mathbb{Z}}}_2$

Thus, the required elementary divisors are {2,2,2,2} and invariant factors are: {2,2,2,2}.