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text: '[summary: The cyclic [3gd groups] are the simplest kind of group; they are the groups which can be made by simply "repeating a single element many times". For example, the rotations of a polygon.]\n\n[summary(technical): A [3gd group] $G$ is **cyclic** if it has a single [generator_mathematics generator]: there is one [element_mathematics element] $g$ such that every element of the group is a [ power] of $g$.]\n\n# Definition\n\nA cyclic group is a group $(G, +)$ (hereafter abbreviated as simply $G$) with a single generator, in the sense that there is some $g \\in G$ such that for every $h \\in G$, there is $n \\in \\mathbb{Z}$ such that $h = g^n$, where we have written $g^n$ for $g + g + \\dots + g$ (with $n$ terms in the summand).\nThat is, "there is some element such that the group has nothing in it except powers of that element".\n\nWe may write $G = \\langle g \\rangle$ if $g$ is a generator of $G$.\n\n# Examples\n\n - $(\\mathbb{Z}, +) = \\langle 1 \\rangle = \\langle -1 \\rangle$\n - The group with two elements (say $\\{ e, g \\}$ with identity $e$ with the only possible group operation $g^2 = e$) is cyclic: it is generated by the non-identity element. Note that there is no requirement that the powers of $g$ be distinct: in this case, $g^2 = g^0 = e$.\n - The integers [modular_arithmetic modulo] $n$ form a cyclic group under addition, for any $n$: it is generated by $1$ (or, indeed, by $n-1$).\n - The [497 symmetric groups] $S_n$ for $n > 2$ are *not* cyclic. This can be deduced from the fact that they are not [3h2 abelian] (see below).\n\n# Properties\n\n## Cyclic groups are [3h2 abelian]\nSuppose $a, b \\in G$, and let $g$ be a generator of $G$. Suppose $a = g^i, b = g^j$. Then $ab = g^i g^j = g^{i+j} = g^{j+i} = g^j g^i = ba$.\n\n## Cyclic groups are [2w0 countable]\nThe elements of a cyclic group are nothing more nor less than $\\{ g^0, g^1, g^{-1}, g^2, g^{-2}, \\dots \\}$, which is an enumeration of the group (possibly with repeats).',
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