Why a group is not a direct or semi direct product

[DeldotB]

Homework Statement

Good day all!

(p.s I don't know why every time I type latex [ tex ] ... [ / tex ] a new line is started..sorry for this being so "spread" out)

So I was wondering if my understanding of this is correct:

The Question asks: "$$\mathbb{Z}_4$$ has a subgroup is isomorphic to $$\mathbb{Z}_2$$ The quotient $$\mathbb{Z}_4/ \mathbb{Z}_2$$ is also isomorphic to $$\mathbb{Z}_2$$Nevertheless, $$\mathbb{Z}_4$$ is not a direct or semidirect product of two copies of $$\mathbb{Z}_2$$. Why?

Homework Equations

None

The Attempt at a Solution

Well: Its pretty easy to see that $$\mathbb{Z}_{4}\neq \mathbb{Z}_{2}\oplus \mathbb{Z}_{2}$$ because 2 is not relatively prime to 2. Thus, Z4 isn't a direct product.

For the semi direct product: since $$\mathbb{Z}_{2}\cap \mathbb{Z}_{2}\neq e$$ (where e is the identity), Z4 is not a semi direct produt of two copies of Z2. Is this correct? (and sufficient?)

 

[Orodruin]

(p.s I don't know why every time I type latex [ tex ] ... [ / tex ] a new line is started..sorry for this being so "spread" out)

Because this is the definition of that tag. Use [ itex] or double hashes instead if you want to write equations in text.

$\mathbb Z_4$ is not isomorphic to $\mathbb Z_2$. You cannot have a bijection between sets of different cardinality.

Can you think of a property of any element in $\mathbb Z_4$ which none of the products reproduce?

 

[DeldotB]

Because this is the definition of that tag. Use [ itex] or double hashes instead if you want to write equations in text.

$\mathbb Z_4$ is not isomorphic to $\mathbb Z_2$. You cannot have a bijection between sets of different cardinality.

Can you think of a property of any element in $\mathbb Z_4$ which none of the products reproduce?

[1] in Z4 has order 4

 

[DeldotB]

<1>

Because this is the definition of that tag. Use [ itex] or double hashes instead if you want to write equations in text.

$\mathbb Z_4$ is not isomorphic to $\mathbb Z_2$. You cannot have a bijection between sets of different cardinality.

Can you think of a property of any element in $\mathbb Z_4$ which none of the products reproduce?

I miss typed the question. I meant to say Z4 has a subgroup isomorphic to Z2 (obviously subgroup generated by [2])