Proof: any element of a group of prime order generates the group

The following self-contained post clarifies the proof of theorem 4, chapter 13, p. 129, of the book “A book of abstract algebra” by Charles C. Pinter.

Let G be a group of prime order |G|=p. It will be shown that for any a\in G with a\ne e it holds that \langle a\rangle=G, where \langle a\rangle is the set generated by a and e is the identity element of G.

Since the order of group G is |G|=p<\infty, the order \mbox{ord}(a) of element a is also finite, i.e. \mbox{ord}(a)=n<\infty for some positive integer n.

Note that the set \langle a\rangle=\left\{a^0,a^1,\ldots,a^{n-1}\right\} generated by a is a subgroup of G, since \langle a\rangle is closed with respect to multiplication and with respect to inverses. Indeed, for any a^i,a^j\in\langle a\rangle, the division algorithm of integers ensures that there exist integers q,r with 0\le r <n such that a^i a^j = a^{i+j}=a^{nq+r}=(a^n)^q a^r=e a^r=a^r\in \langle a \rangle. Moreover, the inverse of any a^i\in \langle a \rangle is (a^i)^{-1}=a^{n-i}\in \langle a \rangle.

The order |\langle a \rangle| of the cyclic subgroup \langle a \rangle of G is equal to the order of generator a, that is |\langle a \rangle|=\mbox{ord}(a)=n.

By Lagrange’s theorem, n is a factor of p. Since p is a prime, n=1 or n=p. It follows from a\ne e that n\ne 1, so n=p, which means that |\langle a \rangle|=|G|.

\langle a \rangle\subseteq G and |\langle a \rangle|=|G|\Rightarrow\langle a \rangle=G.

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