868: Left or Right Coset of Subgroup by Element of Group

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definition of left or right coset of subgroup by element of group

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Topics

About: group

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The table of contents of this article

• Starting Context
• Target Context
• Orientation
• Main Body
• 1: Structured Description
• 2: Note

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Starting Context

• The reader knows a definition of group.

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Target Context

• The reader will have a definition of left or right coset of subgroup by element of group.

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Orientation

There is a list of definitions discussed so far in this site.

There is a list of propositions discussed so far in this site.

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Main Body

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1: Structured Description

Here is the rules of Structured Description.

Entities:
$G^{\prime }$: $\in \{\text{ the groups }\}$
$G$: $\in \{\text{ the subgroups of }{G}^{\prime }\}$
$g^{\prime }$: $\in G^{\prime }$
$\ast g^{\prime }G$: $=\text{ the left coset of }G\text{ by }{g}^{\prime }$
$\ast G{g}^{\prime }$: $=\text{ the right coset of }G\text{ by }{g}^{\prime }$
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Conditions:
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2: Note

For any $g_1^{\prime },g_2^{\prime }\in G^{\prime }$, $g_1^{\prime }G\cap g_2^{\prime }G=\mathrm{\varnothing }$ or $g_1^{\prime }G=g_2^{\prime }G$: when $g_1^{\prime }G\cap g_2^{\prime }G\ne \mathrm{\varnothing }$, there is a $g_3^{\prime }\in g_1^{\prime }G\cap g_2^{\prime }G$, and $g_1^{\prime }G=g_3^{\prime }G=g_2^{\prime }G$, by the proposition that with respect to any subgroup, the coset by any element of the group equals a coset if and only if the element is a member of the latter coset, whether they are left cosets or right cosets.

Likewise, for any $g_1^{\prime },g_2^{\prime }\in G^{\prime }$, $G{g}_1^{\prime }\cap G{g}_2^{\prime }=\mathrm{\varnothing }$ or $G{g}_1^{\prime }=G{g}_2^{\prime }$.

$|g^{\prime }G|=|G|$: for each $g_1,g_2\in G$ such that $g_1\ne g_2$, $g^{\prime }{g}_1\ne g^{\prime }{g}_2$, because supposing that $g^{\prime }{g}_1=g^{\prime }{g}_2$, $g_1={g^{\prime }}^{-1}{g}^{\prime }{g}_1={g^{\prime }}^{-1}{g}^{\prime }{g}_2=g_2$, a contradiction.

Likewise, $|G{g}^{\prime }|=|G|$.

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References

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