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

Loading Question

asked in Set Theory & Algebra by Veteran (10.8k points)   | 160 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 (25.2k 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 Sep 2017
  1. Habibkhan

    6334 Points

  2. Warrior

    2202 Points

  3. Arjun

    2150 Points

  4. nikunj

    1980 Points

  5. manu00x

    1726 Points

  6. SiddharthMahapatra

    1718 Points

  7. Bikram

    1706 Points

  8. makhdoom ghaya

    1650 Points

  9. A_i_$_h

    1518 Points

  10. rishu_darkshadow

    1512 Points


25,979 questions
33,554 answers
79,344 comments
31,011 users