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text: '[summary: A presentation $\\langle X \\mid R \\rangle$ of a [3gd group] is, informally, a way of specifying the group by a set $X$ of *generators* together with a set $R$ of *relators*.\nEvery element of the group is some product of generators, and the relators tell us when a product is trivial.]\n\n[summary(Technical): A presentation $\\langle X \\mid R \\rangle$ of a [3gd group] $G$ is a set $X$ of *generators* and a set $R$ of *relators* which are words on $X \\cup X^{-1}$, such that $G \\cong F(X) / \\llangle R \\rrangle^{F(X)}$ the [-normal_closure] of $\\llangle R \\rrangle$ with respect to the [-5kg] $F(X)$. ]\n\nA presentation $\\langle X \\mid R \\rangle$ of a group $G$ is an object that can be viewed in two ways:\n\n - a way of making $G$ as a [4tq quotient] of the [-5kg] on some set $X$\n - a set $X$ of *generators* (from which we form the set $X^{-1}$ of formal inverses to $X$%%note:For example, if $X = \\{ a, b \\}$ then $X^{-1}$ is a set of new symbols which we may as well write $\\{ a^{-1}, b^{-1} \\}$. %%), and a set $R$ of *relators* (which must be [5jc freely reduced words] on $X \\cup X^{-1}$), such that every element of $G$ may be written as a product of the generators and such that by combining elements of $R$ we can obtain every possible way of expressing the identity element of $G$ as a product of elements of $X$.\n\nEvery group $G$ has a presentation with $G$ as the set of generators, and the set of relators is the set containing every trivial word.\nOf course, this presentation is in general not unique: we may, for instance, add a new generator $t$ and the relator $t$ to any presentation to obtain an [49x isomorphic] presentation.\n\nThe above presentation corresponds to taking the quotient of the free group $F(G)$ on $G$ by the homomorphism $\\phi: F(G) \\to G$ which sends a word $(a_1, a_2, \\dots, a_n)$ to the product $a_1 a_2 \\dots a_n$.\nThis is an instance of the more widely-useful fact that every group is a [4tq quotient] of a [-5kg] ([5jb proof]).\n\n# Examples\n\n- The [-47y] $C_2$ on two elements has a presentation $\\langle x \\mid x^2 \\rangle$. That is, it has just one generator, $x$, and the relator $x^2$ tells us that $x^2$ is the [54p identity] $e$.\nNotice that $\\langle x \\mid x^4 \\rangle$ would also satisfy the description that "there is one generator, and $x^4$ is the identity". However, the group corresponding to *this* presentation contains four elements, not two, so it is not $C_2$. This demonstrates the fact that if we have a presentation $\\langle X \\mid R \\rangle$, and a group can be written in such a way that all the relators hold in the group, and the group can be generated by the elements of $X$, that still doesn't mean the presentation describes the group; it could be that extra relations hold in the group that aren't listed in $R$. (In this case, for example, $x^2 = e$ is not listed in $\\langle x \\mid x^4 \\rangle$.)\n- The presentation $\\langle x, y \\mid xyx^{-1}y^{-1} \\rangle$ describes a group with two generators, such that the only nontrivial relation is $xyx^{-1}y^{-1} = e$ (and anything that can be built up from that). That relation may be written as $xy=yx$: that is, $x$ and $y$ [3jb commute]. This tells us that the group is [3h2 abelian], since every generator commutes with every other generator. In fact, this group's elements are just words $x^n y^m$ for some integers $n, m$; this follows because, for instance, $xyx = xxy = x^2y$, and in general we can pull all the instances of the letter $x$ (and $x^{-1}$) out to the front of the word. Therefore we can write an element of this group as $(m, n)$ where $m, n$ are integers; hence the group is just $\\mathbb{Z}^2$ with pointwise addition as its operation.\n- The presentation $\\langle x, y \\mid x^2, y \\rangle$ is just $C_2$ again. Indeed, we have a relator telling us that $y$ is equal to the identity, so we might as well just omit it from the generating set (because it doesn't add anything new to any word in which it appears).\n- The presentation $\\langle a, b \\mid aba^{-1}b^{-2}, bab^{-1}a^{-2} \\rangle$ is a longwinded way to define the trivial group (the group with one element). To prove this, it is enough to show that each generator represents the identity, because then every word on the generators has been made up from the identity element so is itself the identity. We have access to the facts that $ab = b^2 a$ and that $ba = a^2 b$ in this group (because, for example, $aba^{-1} b^{-2} = e$). The rest of the proof is an exercise.\n\n%%hidden(Show solution):\nWe have $ab = b^2 a$ from the first relator; that is $b ba$.\nBut $ba = a^2 b$ is the second relator, so that is $b a^2 b$; hence $ab = b a^2 b$ and so $a = b a^2$ by cancelling the rightmost $b$.\nThen by cancelling the rightmost $a$, we obtain $e = ba$, and hence $a = b^{-1}$.\n\nBut now by the first relator, $ab = b^2 a = b b a$; using that both $ab$ and $ba$ are the identity, this tells us that $e = b$; so $b$ is trivial.\n\nNow $a = b^{-1}$ and so $a$ is trivial too.\n%%\n\n\n\n[todo: finite presentation/generation]\n[todo: direct products]\n[todo: semidirect products]',
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