document.write( "Question 1045096: what is a group? And construct three example of groups. \n" ); document.write( "
Algebra.Com's Answer #660498 by ikleyn(52946)\"\" \"About 
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\n" ); document.write( "what is a group? And construct three example of groups.
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\n" ); document.write( "\n" ); document.write( "Examples of groups:\r
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document.write( "1.  All integer numbers with the operation \"adding\" on them.\r\n" );
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document.write( "2.  All real numbers with the operation \"adding\" on them.\r\n" );
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document.write( "3.  All real numbers except zero with the operation \"multiplication\" on them.\r\n" );
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document.write( "4.  All complex numbers with the operation \"adding\" on them.\r\n" );
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document.write( "5.  All complex numbers except zero with the operation \"multiplication\" on them.\r\n" );
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\n" ); document.write( "\n" ); document.write( "From Wikipedia ( https://en.wikipedia.org/wiki/Group_(mathematics) )\r
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document.write( "In mathematics, a group is an algebraic structure consisting of a set of elements equipped with an operation \r\n" );
document.write( "that combines any two elements to form a third element. The operation satisfies four conditions called the group axioms, \r\n" );
document.write( "namely closure, associativity, identity and invertibility. \r\n" );
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document.write( "One of the most familiar examples of a group is the set of integers together with the addition operation, \r\n" );
document.write( "but the abstract formalization of the group axioms, detached as it is from the concrete nature of any particular group \r\n" );
document.write( "and its operation, applies much more widely. It allows entities with highly diverse mathematical origins in abstract \r\n" );
document.write( "algebra and beyond to be handled in a flexible way while retaining their essential structural aspects. \r\n" );
document.write( "The ubiquity of groups in numerous areas within and outside mathematics makes them a central organizing principle of contemporary mathematics.\r\n" );
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document.write( "A group is an algebraic structure (G,∘) which satisfies the following four conditions:   ( https://proofwiki.org/wiki/Definition:Group_Axioms )\r\n" );
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document.write( "(G0)  	:  	Closure   	  ∀a,b∈G:	a∘b∈G   				         \r\n" );
document.write( "(G1)  	:  	Associativity     ∀a,b,c∈G:	a∘(b∘c)=(a∘b)∘c   				         \r\n" );
document.write( "(G2)  	:  	Identity   	  ∃e∈G:∀a∈G:	e∘a=a=a∘e   				         \r\n" );
document.write( "(G3)  	:  	Inverse   	  ∀a∈G:∃b∈G:	a∘b=e=b∘a   				         \r\n" );
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