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Prabhupada posted an Question
April 01, 2020 • 23:48 pm
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30 points
30 points
- IIT JAM
- Mathematics (MA)
Show that there doesn't exist any element of order 2 in a group of odd order
show that there doesn't exist any element of order 2 in a group of odd order
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Priyadarshan Choursiya Best Answer
let |G| = 2n+1 for some n€ N. suppose there exist g€G such that |g| = 2. then order of cyclic group || = |g| = 2.
where = {e,g}
by Lagrange's thm, every subgroup divides order of group, so (2n+1)mod2 =0, which is not possible, since 2 doesn't divide odd no.
hence there doesn't exist such g.
therefore G doesn't have any element of order 2.
the proof of Lagrange's thm you understand when you know about costes.