| Lagrange's Theorem (group Theory) |
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Lagrange's theorem, in the Mathematics of Group Theory , states that if ''G'' is a finite Group and ''H'' is a Subgroup of ''G'', then the Order (that is, the number of elements) of ''H'' Divides the order of ''G''. It is named after Joseph Lagrange . This can be shown using the concept of left because its inverse is given by ''f'' -1(''y'') = ''ab''-1''y''. | ||
| :''G'' | <b> {Link without Title} </b> · ''H'', |
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| The Converse Of Lagrange's Theorem Is Not True In General: Given A Finite Group ''G'' And A Divisor ''d'' Of ''G'', There Does Not Necessarily Exist A Subgroup Of ''G'' With Order ''d'' The Smallest Example Is The | "http://wwwinformationdelightinfo/encyclopedia/entry/alternating_group" class="copylinks">Alternating Group ''G'' = ''A''<sub>4</sup> which has 12 elements but no subgroup of order 6 However, if ''G'' is Abelian , then there always exists a subgroup of order ''d'' A partial converse for the general case is given by Cauchy's Theorem |
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