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I am asked Is $3$ a generator of $(Z_5;+$5$)$? What about $2, 1,$ and $4$?

I found that

$3^1 = 3 % 5 = 3$

$3^2 = 9 % 5 = 4$

$3^3 = 27 % 5 = 2$

$3^4 = 81 % 5 = 1 $

$3^5 = 243 % 5 = 3$

Thus i found it isn't a generator. However, the answer indicates that it is. More specificaly that

$3^1= 3 $

$3^2= 1 $

$3^3= 4$

$3^4= 2$

$3^5= 0$

Could anyone explain why this is? I am confused.

  • 4
    You were given that the operation was addition, but you used multiplication. – 2017-01-18
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    So for $3^3$, I should be doing $3+3+3$, not $3*3*3$? – 2017-01-18
  • 3
    Right, though using the notation $3^3$ is usually not a good idea when the operation is addition. – 2017-01-18
  • 0
    Yes it can causes confusion. Thanks for the help! – 2017-01-18
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    In group theory you need to be able to handle either $+$-based or $\times$-based notation. OTOH, it _is_ customary to use the one that's more natural for the problem domain. – 2017-01-18

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