document.write( "Question 1029391: What is Subgroup
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Algebra.Com's Answer #644447 by ikleyn(52787) $\"\"$ $\"About$

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document.write( "A subgroup of a group is a subset of the group  a) closed relative to the group operation,  and  \r\n" );
document.write( "                                                b) closed relative taking the opposite (reciprocal) elements.\r\n" );
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document.write( "So, if  G  is a group and  H  is a subgroup in  G,  then  H is a subset of  G  and for any two elements  \"a\"  and  \"b\"  of  H  \r\n" );
document.write( "their product  a*b  belongs to  H,  and for any element  \"a\"  of  H  its opposite  -a  (or its reciprocal  )  belongs to H.\r\n" );
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document.write( "Examples.\r\n" );
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document.write( "All even integer numbers form the subgroup in the group of all integers for addition.\r\n" );
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document.write( "All integers multiple 3 form the subgroup in the group of all integers for addition.\r\n" );
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document.write( "All integer numbers form the subgroup in the group of all real numbers for addition.\r\n" );
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document.write( "All complex numbers with the modulus 1,  {z of C| |z| = 1},  form the subgroup in the multiplicative group of all complex non-zero numbers \r\n" );
document.write( "    {z of C| z =/= 0}  for multiplication.\r\n" );
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document.write( "In accordance with the definition, you should check that\r\n" );
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document.write( "   a) it is a subset in the given group;\r\n" );
document.write( "   b) for any two elements \"a\" and \"b\" of the subset their product (sum, composition) belongs to the subset;\r\n" );
document.write( "   c) for any element \"a\" of the subset its opposite \"-a\" (or ) belongs to the subset.\r\n" );
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document.write( "The simplest way to generate a subgroup is to take ANY element g of the group and collect all elements {ng}  (or {g^n}), \r\n" );
document.write( "       n = 0, +/-1, +/-2, . . . for all integer n. \r\n" );
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document.write( "Notice that the element  (or ) is the zero (neutral, or unit) element of the group. It must be present in any subgroup.\r\n" );
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document.write( "The group described in this section, is minimal subgroup generated by the element \"g\" of G.\r\n" );
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document.write( "You can create larger subgroups by generating them using any two, three . . . elements of the group and taking all their linear combinations/compositions.\r\n" );
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\n" ); document.write( "\n" ); document.write( "One more notice.\r
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\n" ); document.write( "\n" ); document.write( "There are two major types of groups: abelian groups where the group operation is commutative,
\n" ); document.write( "and non-abelian, where the group operation is non-commutative.\r
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\n" ); document.write( "\n" ); document.write( "Traditionally, the group operation in abelian groups is called \"addition\".\r
\n" ); document.write( "\n" ); document.write( "The group operation in non-abelian groups is called \"multiplication\" or \"composition\".\r
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