Identify isomorphism type for each proper subgroup of (Z/32Z)*

[i_a_n]

What is the order of each of the elements of $<\bar{15},\bar{17},\bar{31}>$?
And what is the order of each of the elements of $Z_4$?

Basically isomorphic means that all elements behave the same, have the same order, and have the same multiplication table.

So $<\bar{15},\bar{17},\bar{31}>$ is isomorphic to $Z_4\times Z_4$? since every element in $Z_4\times Z_4$ has order 2.

Can you tell me exactly what $<\bar{15},\bar{17},\bar{31}>$ is isomorphic to?

 

[I like Serena]

So $<\bar{15},\bar{17},\bar{31}>$ is isomorphic to $Z_4\times Z_4$? since every element in $Z_4\times Z_4$ has order 2.

It can't be, since $Z_4\times Z_4$ has order 16.

Can you tell me exactly what $<\bar{15},\bar{17},\bar{31}>$ is isomorphic to?

Okay, okay, it's isomorphic to $Z_2 \times Z_2$.
$\bar{1} \mapsto (0,0)$
$\bar{15} \mapsto (0,1)$
$\bar{17} \mapsto (1,0)$
$\bar{31} \mapsto (1,1)$

This matches since $\bar{15} \cdot \bar{15} = \bar 1$, while at the same time $(0,1) + (0,1) = (0,0) \mod 2$.
And we have $\bar{15} \cdot \bar{17} = \bar{31}$, while at the same time $(0,1) + (1,0) = (1,1) \mod 2$.

 

[i_a_n]

Thank you