Does it make any sense to raise an element in a group $G$ to the power of any other element in the same group? E.g $a^b=?$ where $a,b \in G$. (multiplying $a$ $b$ times, but $b$ can or can't be a number)
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It doesn't, in general. But you're dealing with abstract mathematics. You can yourself define this to be some sought of operation or composition law (let's say) or any abstract non-sense which you may like. – kishlaya Feb 04 '17 at 12:51
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2$a^b$ is often used as a shorthand for the conjugate $b^{-1}ab$. Note that then we have the "exponent law" $a^{bc} = (a^b)^c$, which may explain the use of the notation. – Daniel Fischer Feb 04 '17 at 12:57
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It does not make sense in general.
What would $a^b$ means if $a$ and $b$ are any matrix of $M_n(\mathbb C)$ for example?
Or if $a$ and $b$ are elements of the group of symmetries of a molecule?
E. Joseph
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@I.R You are right, but this only work for some matrix. And $(M_n(\mathbb C),+)$ is a group, so you can not generally define $M^N$ for two matrix $M$ and $N$. – E. Joseph Feb 04 '17 at 12:54