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 G on a set A is a map from G×A→A usually written as g⋅a for all g∈G and a∈A, that satisfies the following two properties:
- g1⋅(g2⋅a)=(g1g2)⋅a for all g1,g2∈G,a∈A
