Difference between revisions of "Group action"
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==Definition== | ==Definition== | ||
A ''group action'' of a group {{M|G}} on a set {{M|A}} is a map from {{M|G\times A \to A}} usually written as {{M|g\cdot a}} for all {{M|g\in G}} and {{M|a\in A}}, that satisfies the following two properties: | A ''group action'' of a group {{M|G}} on a set {{M|A}} is a map from {{M|G\times A \to A}} usually written as {{M|g\cdot a}} for all {{M|g\in G}} and {{M|a\in A}}, that satisfies the following two properties: | ||
| − | * {{M|g_1 \cdot(g_2\cdot a) =(g_1g_2)\cdot a}} for all {{M|g_1,g_2\in G,a\in A}} | + | * {{M|1=g_1 \cdot(g_2\cdot a) =(g_1g_2)\cdot a}} for all {{M|g_1,g_2\in G,a\in A}} |
| − | {{Definition|Abstract Algebra | + | {{Definition|Abstract Algebra|Group Theory}} |
| − | + | ||
Revision as of 10:54, 20 February 2016
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Definition
A group action of a group [ilmath]G[/ilmath] on a set [ilmath]A[/ilmath] is a map from [ilmath]G\times A \to A[/ilmath] usually written as [ilmath]g\cdot a[/ilmath] for all [ilmath]g\in G[/ilmath] and [ilmath]a\in A[/ilmath], that satisfies the following two properties:
- [ilmath]g_1 \cdot(g_2\cdot a) =(g_1g_2)\cdot a[/ilmath] for all [ilmath]g_1,g_2\in G,a\in A[/ilmath]
