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The order of the smallest possible non trivial group containing elements $x$ and $y$ such that $x^7 = y^2 = e $ and $ yx = x^4 y$ is

  1. 1
  2. 2
  3. 7
  4. 14

I am stuck on this problem. Can anyone help me please?

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Are these questions from Subject GRE? –  Ram Jan 2 '13 at 6:33
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Should we assume both $x$ and $y$ are nontrivial? –  anon Jan 2 '13 at 6:36
    
GRE means,????? –  Prasanta Jan 2 '13 at 6:36
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I'm with @anon on this. Any group contains such elements $x$ and $y$, because we can always take $x=y=1$. Of course, in some groups there can be other suitable pairs $(x,y)$, but this is irrelevant if the question is formulated as it is. So some kind of an additional non-triviality assumption is required to make the question at least a bit interesting. –  Dan Shved Jan 2 '13 at 7:01
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4 Answers 4

up vote 3 down vote accepted

Hint:

  1. Lagrange's theorem: For any finite group $G$, the order of every subgroup $H$ of $G$ divides the order of $G$.

  2. $yx = x^4y$; $y(yx) = x$ do the manipulations using $yx = x^4y$ again and again and you will get $x = x^m$ for some $m \le 7 $ use that fact

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assume $y \neq 0$ $yx = x^4y$ left multiply both sides by $y$ $ x = y x^4y$ $\Rightarrow $ $x = x^4yx^3y$ $\Rightarrow$ $ x = x^4x^4yx^2y$ finally you will get some relation between powers of $x$, you can use that. –  Ram Jan 2 '13 at 6:51
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@ Ram, I get $x=x^{16}$,so,$x=x^2$,so $x=e$....which is wrong?? –  Prasanta Jan 2 '13 at 6:59
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If x = e, plug it back into the relations you are given. Can you find out anything new about y? –  Billy Jan 2 '13 at 7:07
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It would be helpful for me to improve my answers if reason for down vote is provided. I gave this particular method as answer since these type of questions appear very frequently in Indian exams and I felt it would be nice if I give a general procedure. –  Ram Jan 2 '13 at 7:15
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@Ram I downvoted because I feel that this is not the best approach to the question. There's no need to prove that $x=e$ when we can simply set it to equal $e$. Although I do agree that proving it can also improve one's understanding of the problem. –  Dan Shved Jan 2 '13 at 7:21
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For any group of order $2, say~\{e,a\}$ taking $x=e,y=a$ satisfies both the equation.

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First solution - $x=y=e$ satisfies the relations. The smallest non-trivial group has order 2, and the relations can be satisfied within that group.

Now suppose we want $x$ and $y$ distinct (not stated in the question).

If $x$ and $y$ are both non-trivial (i.e. $\neq e$) then the first relation shows that the group must contain non-trivial elements of orders 2 and 7, and then Lagrange means that the order of such a group must be divisible by 14. [note we have not used the second relation or shown it is compatible with this conclusion]

So to get a non-trivial group of order less than 14, one of $x$ or $y$ must be the identity. If we set $y=e$ we see that $x^7=x^3=1$ so that $x=e$, which is not what we want. If we set $x=e$ then $y^2=e$ and this can be done in a group of order 2.

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Up to isomorphism, there is only one group of order 1, one group of order 2, one group of order 7, and two groups of order 14. Figure out (or look up) what those groups are. Which ones are non-trivial? Look for suitable elements $x$ and $y$ in those groups. You may want to start from the smallest group and work your way up.

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Nice answer, but I doubt that might be little advanced for him right now. –  Ram Jan 2 '13 at 6:56
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