40 lines
1 KiB
Markdown
40 lines
1 KiB
Markdown
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---
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title: groups
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tags: maths, groups
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---
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# Quotient groups
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Wiki: https://en.wikipedia.org/wiki/Coset
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`Normal` subgroup of **G** is any subgroup of **G** which is closed with respect to conjugates.
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### Theorem 1
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If $H$ is a normal subgroup of $G$, then $aH = Ha$ for every $a \in G$.
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There is no distinction between left and right cosets for a normal subgroup.
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### Theorem 2
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Let $H$ be a normal subgroup of $G$. If $Ha = Hc$ and $Hb = Hd$, then $H(ab) = H(cd)$.
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### Theorem 3
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$G/H$ with coset multiplication is a group.
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### Theorem 4
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$G/H$ is a homomorphic image of $G$.
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### Theorem 5
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Let $G$ be a group and $H$ a subgroup of $G$. Then
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- $Ha = Hb$ _iff_ $ab^{-1} \in H$ and
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- $Ha = H$ _iff_ $a \in H$
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The motive for the quotient group construction is that it gives a way of actually _producing_ all the homomorphic images of any group $G$.
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For quotient group construction in practical instances, $H$ is often chosen to "factor out" unwanted properties of $G$, and preserve in $G/H$ only "desirable" traits.
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