Does someone know the why raising the element of a group to the power of the order of the group yields the identity?
By (finite) group I mean a tuple (G,*) that satisfies the following:
- closure under operation *
- associative under operation *
- has identity element
- has inverse
I was unsure why:
$$
a^{|G|} = I
$$
[I am aware that its a basic fact but I was unable to find the proof online [don't know the name of the theorem]). I wanted to try to prove it myself, but I was not sure how to because, the statement of the theorem does not seem to state anything about what is allowed to be an element of the group and what isn't allowed, does it only apply to integers? What is allowed to be an element of the group (binary strings?)? Also, I was not sure what a raised to the order meant, because, the group might not have "multiplication" defined, maybe it has its own way of doing stuff to its elements, so I didn't know what raising it to the power of meant. Does it mean using operation * |G| times?
Does this only apply to cyclic groups?
I usually post my (failed) attempts to solve the question first but I was not sure how to start proving anything because I was not sure what the elements of the group were allowed to be or what the raising meant with respect to operations on the elements of the group.
Can anyone state the name of the theorem?
Best Answer
If $G$ has $|G|$ many elements, then the set $$\{a, a^2, a^3, \cdots, a^{|G|}\}$$
either has a repetition or exhaust the group. If it exhaust the group, one of these elements is $1$. If there is a repetition, so $a^{r} = a^{s}$ for $r>s$, then $a^{r-s} = 1$. In any event, at least one of these elements must equal $1$, say $a^n = 1$, and choose $n$ to be the smallest such exponent. If we can show that $n$ divides $|G|$, we are done. But $n$ is the order of the subgroup $$\{a, a^2, \cdots , a^n(=1)\}$$
And by Lagrange's theorem, the order of a subgroup must divide the order of the group.
To rephrase here a bit: If $a$ is a generator of $G$, then raising it to the power of the order of the group guarantees that it will cycle through all the elements and return to the identity. If $a$ is not a generator of $G$ then Lagrange's theorem guarantees that the order of the subgroup generated by $a$ divides $|G|$ and therefore if $a^{|a|}=1$ then $(a^{|a|})^{u}=1$ where $|G|=u|a|$.
In response to your questions about definitions: A group is an abstract object. We do not know anything about the sorts of objects inside the group. All we know about the group is that it satisfies certain axioms and has a (binary) group operation $\ast$. Given a pair of elements $a,b \in G$, we write $a \ast b$ or $ab$ to denote the group operation acting on the pair of elements. The exponent notation means: $a^{2} = aa, a^{7} = aaaaaaa$.