Why a group is not a direct or semi direct product

  • #1
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Homework Statement


Good day all!

(p.s I dont 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: "[tex]\mathbb{Z}_4[/tex] has a subgroup is isomorphic to [tex]\mathbb{Z}_2[/tex] The quotient [tex]\mathbb{Z}_4/ \mathbb{Z}_2[/tex] is also isomorphic to [tex]\mathbb{Z}_2[/tex]Nevertheless, [tex]\mathbb{Z}_4[/tex] is not a direct or semidirect product of two copies of [tex]\mathbb{Z}_2[/tex]. Why?

Homework Equations


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The Attempt at a Solution


Well: Its pretty easy to see that [tex]\mathbb{Z}_{4}\neq \mathbb{Z}_{2}\oplus \mathbb{Z}_{2} [/tex] becuase 2 is not relatively prime to 2. Thus, Z4 isnt a direct product.

For the semi direct product: since [tex]\mathbb{Z}_{2}\cap \mathbb{Z}_{2}\neq e[/tex] (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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Answers and Replies

  • #2
Orodruin
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(p.s I dont 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?
 
  • #3
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7
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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  • #4
117
7
<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])
 

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