# 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?

None

## The Attempt at a Solution

Well: Its pretty easy to see that $$\mathbb{Z}_{4}\neq \mathbb{Z}_{2}\oplus \mathbb{Z}_{2}$$ becuase 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?)

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(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

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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])