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Why a group is not a direct or semi direct product
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[QUOTE="DeldotB, post: 5248199, member: 562269"] [h2]Homework Statement [/h2] 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: "[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? [h2]Homework Equations[/h2] None [h2]The Attempt at a Solution[/h2] 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 isn't 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?) [/QUOTE]
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Why a group is not a direct or semi direct product
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