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So in abstract algebra the elements in a group do not have to be anything in particular, they can be numbers, matrices, sets, etc, and the operation on them can also be many things, addition, multiplication, modular operations, etc. However, when you want to calculate the order of an element, you have to use exponentiation, in particular. Does this mean that the elements of a group have to be something for which exponentiation is defined? Also, what exponentiation even means is completely different depending on the type of element (exponentiation is different for number, matrices, and sets, and it seems to me like the concepts are related by name alone), so how can exponentiation be generally used to find order of any type of element?

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  • $\begingroup$ In a group written additively, the exponent of $x$ is the least $n\in\Bbb N$ with $nx=0$. Here $nx=x+x+\cdots+x$ ($n$ summands). $\endgroup$ – Angina Seng Apr 28 '17 at 15:53
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    $\begingroup$ We use exponential notation (in noncommutative groups; where 'multiplication' is taken as the group operation), but that's just shorthand; this exponentiation is perfectly well defined in terms of the group operation. (In general, $g^n=g\circ g^{(n-1)}$, if $\circ$ is the group op.) $\endgroup$ – Steven Stadnicki Apr 28 '17 at 15:54
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    $\begingroup$ It is just a notation : $x^n=x\star\dots\star x$, where $x$ is repeated $n$ times, and where $\star$ is th group law. $\endgroup$ – Balloon Apr 28 '17 at 15:54
  • $\begingroup$ If you have a group defined in the first place you automatically have "exponentiation" defined as it is just the group operation repeated an integer number of times. It is often written as a superscript like exponentiation since the multiplicative notation for groups is so common, but it could just as easily have been written as scalar multiplication if you were preferring to use additive notation for groups as well. $\endgroup$ – JMoravitz Apr 28 '17 at 15:54
  • $\begingroup$ In a set with a product, exponentiation to the power of an integer is always defined. It's simply repeated multiplication (or addition, etc.) $\endgroup$ – Matt Samuel Apr 28 '17 at 15:54
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Group is the set $G$ with binary operation $*$ defined on it satisfying group axioms. The exponentiation is defined recursively as:

$x^1 := x$, $x^n := x * x^{n-1}$ for $n>3$.

Similarly for negative exponents. Thus exponentiation is nothing but repeated use of $*$.

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