GATE CSE
First time here? Checkout the FAQ!
x
+6 votes
129 views

Loading Question

asked in Set Theory & Algebra by Veteran (10.5k points)   | 129 views
Is it $31$?
yes, but how???

1 Answer

+9 votes
Best answer

Given that, $a^5 = e$ and $aba^{-1} = b^2$, $a,b \in G$

Order of an element of a group is the smallest $m$ such that $a^m = e$. So, $O(a) = 5$

$a^2ba^{-2} = a(aba^{-1})a^{-1} = ab^2a^{-1} = (aba^{-1})(aba^{-1}) = b^2b^2 = b^4$


$\color{navy}{a^3ba^{-3} = a^2(aba^{-1})a^{-2} \\= a^2(b^2)a^{-2} \\= a(aba^{-1})(aba^{-1})a^{-1} \\= a(b^2b^2)a^{-1} \\= (aba^{-1})(aba^{-1})(aba^{-1})(aba^{-1}) \\= b^2b^2b^2b^2 \\= b^8}$


Similarly, $a^4ba^{-4} = b^{16}$ and $a^5ba^{-5} = b^{32}$

  • In a group inverse of identity element is identity element itself. 

So, $a^5 = e$ and $a^{-5} = e^{-1} = e$ 

Multiplying both sides by $b^{-1}$, we get $\Rightarrow$

$b^{31} = e \Rightarrow \color{red}{O(b) = 31}$

Reference : http://fmwww.bc.edu/gross/MT310/hw02ans.pdf (See example $7$ and $8$)

answered by Veteran (24.7k points)  
selected by
thank you, really nice question and explaination.
Yes, it's really a good question.

Really very nice question and nice explanation as well by u @mcjoshi..The step :

ab2a-1=(aba-1)(aba-1) was the trickiest step in my opinion which did not strike to me.

Thanks Habib.
How is this derived: ab^2a−1=(aba^−1)(aba^−1)

Thanks
$a*a^{-1} = e$(identity element)
Sorry, I still am not able to get how $ab^{2}a^{−1} = (a.b.a^{−1})(a.b.a^{−1})$. Thanks
$a*a^{-1} = e$. So, $ab^2a^{-1}$ can be written as : $ab(a^{-1}a)(a^{-1}a)(a^{-1}a)ba^{-1}$, means writing $(a^{-1}a) $anywhere does not change meaning. You can consider $(a^{-1}a) = 1$
Thank you, that was helpful.


Top Users May 2017
  1. akash.dinkar12

    3308 Points

  2. pawan kumarln

    1884 Points

  3. Bikram

    1656 Points

  4. sh!va

    1640 Points

  5. Arjun

    1396 Points

  6. Devshree Dubey

    1272 Points

  7. Debashish Deka

    1162 Points

  8. Angkit

    1048 Points

  9. LeenSharma

    1010 Points

  10. Arunav Khare

    754 Points

Monthly Topper: Rs. 500 gift card
Top Users 2017 May 22 - 28
  1. Bikram

    742 Points

  2. pawan kumarln

    510 Points

  3. Arnab Bhadra

    490 Points

  4. bharti

    304 Points

  5. LeenSharma

    248 Points


22,832 questions
29,158 answers
65,233 comments
27,673 users