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Antima Jangid posted an Question
November 23, 2021 • 15:38 pm
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30 points
30 points
- IIT JAM
- Mathematics (MA)
145. there exists a finite abelian group g containing exactly 60 elements of order 2. (t/f) (tifr 2017)
145. There exists a finite abelian group G containing exactly 60 elements of order 2. (T/F) (TIFR 2017)
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Anonymous User Best Answer
Suppose the order of G is 2n. Since order of identity is always 1 i.e. |e|=1. So we have left only2n−1 element which is odd in number. Elements which are not their own inverses, these elements and their inverses exist in pairs. Then it should be even in number but we have odd number here. In that case we must have atleast an element which must be self inverse. Elements of order 2 are exactly those elements which are self inverse. Thus number of element of order 2 must be in odd number. But we have given 60 number of elements of 2 which is absurd.
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Anonymous User
we know O(a)|O(G) if G contain 2 order element then order of G is even. we know order of identity is one remaining elements order more then one. if G contain 2 order element then number of these elements are odd so no group contain 2 order element