# Subgroups ## Subgroup Constructor **Constructor:** **sub**$\langle\ G\ |\ L\ \rangle ~$: Construct the subgroup $H$ of the fp-group $G$ generated by the words specified by the terms of the *generator list* $L = L_1, \ldots, L_r$. A term $L_i$ of the generator list may consist of a word, a set or sequence of words, a subgroup or a set or sequence of subgroups. The collection of words and groups specified by the list must all belong to the group $G$ and $H$ will be constructed as a subgroup of $G$. **Notes:** **(i)** The generators of $H$ consist of the words specified directly by terms $L_i$ together with the stored generating words for any groups specified by terms of $L_i$. Repetitions of an element and occurrences of the identity element are removed (unless $H$ is trivial). If the `sub`-constructor is invoked with an empty list $L$, the trivial subgroup will be constructed. ### `Example: Subgroups1 (ex-7c2095)` The group $(8, 7\ |\ 2, 3)$ is defined by the presentation $$ < a, b \ | \ a^8, b^7, (ab)^2, (ab^{-1})^3 >, $$ and has a subgroup of index 448 generated by the words $a^2$ and $ab^{-1}$: ```magma > G := FPGroup; > G; Finitely presented group G on 2 generators Relations a^8 = Id(G) b^7 = Id(G) (a * b)^2 = Id(G) (a * b^-1)^3 = Id(G) > H := sub< G | a^2, a * b^-1 >; > H; Finitely presented group H on 2 generators Generators as words in group G x = a^2 y = a * b^-1 ``` ## Coset Enumeration **Intrinsics:** `Index` $(G,~H)$ : The index of $H$ in $G$. `CosetTable` $(G,~H)$ : The coset table for $H$ in $G$ (right action). `Transversal` $(G,~H)$ : A (right) Schreier system of coset representatives for $H$ in $G$ together with the coset map. `SchreierGenerators` $(G,~H)$ : Schreier generators for $H$ as a set of words in $G$. `ToddCoxeter` $(G, H)$ : The Todd–Coxeter procedure for coset enumeration is applied to construct the coset table for the subgroup $H$ of fp-group $G$. The convention here is that $G$ acts on the right. If successful the index of $H$ in $G$ and the coset table are returned otherwise zero is returned for the index. `SetGlobalTCParameters` (~: *parameters*) : Set the parameter values used for indirect invocations of the Todd-Coxeter coset enumeration procedure. `UnsetGlobalTCParameters` (~) : Restore the default values for the parameters used for indirect invocations of the Todd–Coxeter coset enumeration procedure. **Notes:** **(i)** While strictly speaking the user does not need to invoke `ToddCoxeter` explicitly, it underpins many of the algorithms used to compute with subgroups in fp-groups and so a serious user needs to understand how to set the parameters in the case of computations that involve hard coset enumerations. **(ii)** The `ToddCoxeter` intrinsic has a large number of parameters which provide the user with control over the enumeration. Descriptions of these may be found in chapter [Finitely Presented Groups](../FPGroups/index-fpgroups.md#chapgrpfp). Perhaps the most useful are `CosetLimit` which specifies the number of cosets which may be defined before abandoning an enumeration, `Time` which limits the maximum CPU time that can be expended, and `Strategy` which if set to ```Hard"` chooses a strategy which has the best chance of succeeding with a hard enumeration. **(iii)** Two intrinsics, `SetGlobalTCParameters` and `UnsetGlobalTCParameters`, allow a user to set the parameters for implicit calls to the Todd–Coxeter algorithm. **(iv)** Descriptions of the Todd-Coxeter algorithm can be found in Chapter 5 of Holt [[Holt *et al.*, 2005](../../references.md#cite-holt-handbook)] and Chapter 5 of Sims [[Sims, 1994](../../references.md#cite-sims-fpgroups)]. ### `Example: Cosets (ex-476bb1)` These intrinsics are illustrated using the infinite group $G$ defined by $$ x^2 = y^3 = (xy)^7 = 1 $$ and $H$ a certain subgroup of $G$ having index 15. ```magma > G := FPGroup< x, y | x^2, y^3, (x*y)^7 >; > G; Finitely presented group G on 2 generators Relations x^2 = Id(G) y^3 = Id(G) (x * y)^7 = Id(G) > > H := sub< G | x, y*x*y*x*y^-1*x*y^-1, y^-1*x*y^-1*x*y*x*y>; > Index(G, H); 15 > > CosetTable(G, H); Mapping from: Cartesian Product<{ 1 .. 15 }, GrpFP: G> to { 1 .. 15 } $1 $2 -$2 1. 1 2 3 2. 4 3 1 3. 5 1 2 4. 2 6 7 5. 3 8 9 6. 6 7 4 7. 10 4 6 8. 11 9 5 9. 9 5 8 10. 7 11 12 11. 8 12 10 12. 13 10 11 13. 12 14 15 14. 15 15 13 15. 14 13 14 > > T := Transversal(G, H); > T; {@ Id(G), y, y^-1, y*x, y^-1*x, y*x*y, y*x*y^-1, x^y, y^-1*x*y^-1, y*x*y^-1*x, y^-1*x*y*x, y*x*y^-1*x*y^-1, y*x*y^-1*x*y^-1*x, y*x*y^-1*x*y^-1*x*y, y*x*y^-1*x*y^-1*x*y^-1 @} ``` ## Coset Action The columns of the coset table for a subgroup $H$ in fp-group $G$ are permutations giving the action of the generators of $G$ on the cosets of $H$. If these columns are treated as permutation generators of a group $P$ then the homomorphism $\phi : G \rightarrow P$ is an epimorphism of $G$. This quotient group and its kernel often provide useful information about the structure of $G$. In Magma the group $G$ acts on the right of cosets and this should be assumed unless left action is explicitly stated. **Intrinsics:** `CosetAction` $(G, H)$ : The permutation representation $\phi$ of $G$ given by the action of $G$ on the cosets of $H$ in $G$. It returns $\phi$ and the image group $\phi(G)$. If the kernel of $\phi$ can be computed easily then it will also be returned. `CosetImage` $(G, H)$ : The image of the permutation action of $G$ on the cosets of $H$. `CosetKernel` $(G, H)$ : The kernel of the permutation action of $G$ on the cosets of $H$. ### `Example: G23 (ex-6c4e10)` The exceptional group $G_2(3)$ is a homomorph of the fp-group $G$ defined below. First, a permutation representation $G1$ on $351$ points is constructed for $G_2(3)$, and then the subgroup generated by the images of the first four generators of $G$ in $G1$ is computed. (The permutation group intrinsics are described in chapter [Permutation Groups](../../FiniteGroups/PermutationGroups/index-permutation-groups.md#chapgrpperm).) ```magma > F := FreeGroup(7); > z := y*c*a^2*b; > u := x*d; > t := w*c*a*d*b^2*c; > G, psi := > quo< F | a^4, b^4, c^2, d^2, (a,b), (a*c)^2, (b*c)^2, > (c*d)^2, d*a*d*b^-1, y^3, (a^-1*b)^y*d*a^-1*b^-1, > (c*d*a^-1*b)^y*b^-1*a*d*c, a*d*y*d*a^-1*y, x^3, > a^x*b^-1, b^x*a*b, c^x*c, (x*d)^2, (u*z)^6, w^3, > (w,y), (a*b)^w*b^-1*a*d*c, (c*d*a^-1*b)^w*d*c*b^2, > (b^2*c*d)^w*a^-1*b^-1, (c*a^2*b*w)^2, > (a^-1*b)^w*d*a^-1*b^-1, (t*u)^3 >; > z1 := psi(z); > u1 := psi(u); > t1 := psi(t); > H := sub< G | z1*a^2*b^2, u1*c, t1*a^2*b^2 >; > phi, G1, K := CosetAction(G, H); > Degree(G1); 351 > print Order(G1), FactoredOrder(G1); 4245696 [ <2, 6>, <3, 6>, <7, 1>, <13, 1> ] > CompositionFactors(G1); G | G(2, 3) 1 > S := sub< G1 | phi(a), phi(b), phi(c), phi(d) >; > BSGS(S); > S; Permutation group S of degree 351 Order = 64 = 2^6 ``` Thus the images of a, b, c and d in G1 generate the Sylow $2$-subgroup of $G_2(3)$. ## Enumeration of FI-Subgroups If an fp-group $G$ has subgroups of finite index and their index is moderate then they may be enumerated. There are two methods, the first of which enumerates all subgroups down to a given index while the second enumerates normal subgroups down to a given index. **Intrinsics:** `LowIndexSubgroups` $(G, R)$ : Given a group $G$ (possibly the free group), and an expression $R$ defining a positive integer range (see below), determine the conjugacy classes of subgroups of $G$ whose indices lie in the range specified by $R$. The expression $R$ is either an integer $n$ which indicates the range $[1..n]$ or a tuple $\langle m,~n \rangle$ which represents the range $[m..n]$. `LowIndexNormalSubgroups`$(G,~n)$ : The normal subgroups of $G$ having index in the range $[1..n]$ are enumerated. The subgroups are returned as a sequence of records (ordered by subgroup index). The upper bound for $n$ is $10^{10}$. **Notes:** **(i)** The `LowIndexSubgroups` intrinsic has many applications and in favourable situations it can locate subgroups of index several hundred. The intrinsic also has parameters which can be found in chapter [Finitely Presented Groups](../FPGroups/index-fpgroups.md#chapgrpfp). Two useful parameters are: **-** `Limit` which accepts a positive integer giving a limit on the the number of subgroups satisfying expression $R$ that are sought. **-** `Print` which accepts an integer in the range $[0\ \ldots\ 3]$ which controls verbose output. Thus $0$ specifies no verbose output while the integers greater than $0$ turn on increasing amounts of verbose output. **(ii)** Another version of the `LowIndexSubgroups` intrinsic implements what is called a **process**. This version can be restarted and returns each subgroup immediately it is found. If more subgroups are needed then the computation is restarted. This is particularly useful in that a subgroup satisfying the user’s requirements may be found long before all the subgroups specified by $R$ are computed. **(iii)** The algorithm used to find subgroups of small index is described in Section 5.6 of Sims [[Sims, 1994](../../references.md#cite-sims-fpgroups)] and Section 5.4 of Holt [[Holt *et al.*, 2005](../../references.md#cite-holt-handbook)]. ### `Example: Lix1 (ex-c69596)` (Peter Lorimer) The two graphs known as Tutte’s 8-cage and the Conder graph may be constructed as the Cayley graphs of two conjugacy classes of subgroups having index $10$ in the finitely presented group $$ \begin{aligned}.\end{aligned} $$ The low index intrinsic will be used to construct these subgroups. ```magma > G := FPGroup h^3 = a^2 = p^2 = 1, p^h = p, p^a = q, > q^h = r, r^a = s, h^s = h^-1, r^h = p * q * r, > s * r = p * q * r * s, p * q = q * p, > p * r = r * p, p * s = s * p, q * r = r * q, > q * s = s * q>; > LowIndexSubgroups(G, <10, 10>); [ Finitely presented group on 6 generators Index in group G is 10 = 2 * 5 Generators as words in group G $.1 = p $.2 = s $.3 = h $.4 = q^-2 $.5 = a * h^-1 * a * r^-1 $.6 = a * h * a * h^-1 * a * q^-1, Finitely presented group on 6 generators Index in group G is 10 = 2 * 5 Generators as words in group G $.1 = p $.2 = s $.3 = h $.4 = q^-2 $.5 = a * h^-1 * a * r^-1 $.6 = a * h * a * h^-1 * a ] ``` ### `Example: Lix2 (ex-c3d6e8)` A fairly surprising application for the low index subgroup algorithm is the enumeration of the conjugacy classes of subgroups of a finite fp-group. Consider the group $$ {{\operatorname{PGL}}}_2(9) \simeq G = \langle a,b | a^2,\ b^3,\ (ab)^8,\ [a,b]^5,\ [a,(ba)^3b^{-1}]^2 \rangle. $$ ```magma > G := FPGroup< a,b | a^2, b^3, (a*b)^8, (a,b)^5, > (a,(b*a)^3*b^-1)^2 >; > Order(G); 720 ``` In an infinite fp-group, finding all conjugacy classes of subgroups up to index 720 using the low index subgroup algorithm would be extremely difficult. In the case of a fp-group $G$ of order $720$, however, it is not so difficult. ```magma > time sgG := LowIndexSubgroups(G, Order(G)); Time: 31.859 > #sgG; 26 ``` A list of representatives of the 26 conjugacy classes of subgroups of $G$ have been found. For every representative, its index in $G$ and a set of generating words are stored. Here are two examples:- ```magma > sgG[10]; Finitely presented group on 2 generators Index in group G is 60 = 2^2 * 3 * 5 Generators as words in group G $.1 = b $.2 = a * b * a * b * a * b^-1 * a * b * a * b * a * b^-1 * a * b^-1 * a > sgG[21]; Finitely presented group on 1 generator Index in group G is 180 = 2^2 * 3^2 * 5 Generators as words in group G $.1 = (b * a)^2 ``` ## Operations for FI-Subgroups Operations on subgroups that construct a new subgroup or quotient group are listed. Here $G$ will be a finitely presented group with a presentation while $H$ and $K$ will be subgroups having finite index in $G$ unless otherwise stated. The subgroups $H$ and $K$ are assumed to be defined either by generators as words in $G$ or by the coset table for $H$ in $G$. **Intrinsics:** `H^u` : The conjugate $H^u$ where $u$ is a word in $G$. $H$ `meet` $K$ : The intersection of subgroups $H$ and $K$ as a subgroup of $G$. `G / H` : The quotient group of $G$ by the normal closure of subgroup $H$. `Core` $(G,~H)$ : The core of $H$ in $G$. `DerivedSubgroup` $(G)$ : The derived subgroup of $G$. `TorsionFreeRank` $(G)$ : The torsion-free rank of the derived quotient group of $G$. `MaximalOverGroup` $(G,~H)$ : A maximal overgroup of $H$ in $G$. `MinimalOverGroup` $(G,~H)$ : A minimal overgroup of $H$ in $G$. `NormalClosure` $(G,~H)$ : The normal closure of $H$ in $G$. `Normaliser` $(G,~H)$ : The normaliser of $H$ in $G$. **Notes:** **(i)** The first four intrinsics involve using the Todd–Coxeter algorithm. In the case of `DerivedSubgroup` it is necessary for Magma to be able to compute the coset table for the derived group of $G$. Thus the intrinsic will fail if the derived group has very large or infinite index in $G$. **(ii)** If $H$ is defined by a coset table then the `SchreierGenerators` intrinsic can be used to produce generating words for $H$. In many cases this will be done automatically. ### `Example: Subgroups1 (ex-f46c39)` The group $(8, 7\ |\ 2, 3)$ is defined by the presentation $$ < a, b \ | \ a^8, b^7, (ab)^2, (ab^{-1})^3 >, $$ and has a subgroup of index 448 generated by the words $a^2$ and $ab^{-1}$: ```magma > G := FPGroup; > G; Finitely presented group G on 2 generators Relations a^8 = Id(G) b^7 = Id(G) (a * b)^2 = Id(G) (a * b^-1)^3 = Id(G) > H := sub< G | a^2, a * b^-1 >; > H; Finitely presented group H on 2 generators Generators as words in group G x = a^2 y = a * b^-1 ``` The subgroup functions will now be demonstrated on the subgroup $H$. First of all its index and order are calculated. ```magma > Index(G, H); 448 > Order(H); 24 ``` The derived group of $G$ is trivial:- ```magma > DerivedSubgroup(G); Finitely presented group Index in group G is 1 = 1 Subgroup of group G defined by coset table ``` The group $G$ is perfect so it is necessary to look elsewhere to find some other subgroups. ```magma > P := MinimalOvergroup(G, H); > P; Finitely presented group P on 3 generators Index in group G is 224 = 2^5 * 7 Generators as words in group G P.1 = a^2 P.2 = a * b^-1 P.3 = a * b^2 * a^3 * b^-1 * a^3 * b^3 > Order(P); 48 ``` So $H$ sits inside a subgroup $P$ having twice its order. ```magma > N := Normalizer(G, H); > N; Finitely presented group N on 4 generators Index in group G is 112 = 2^4 * 7 Generators as words in group G N.1 = a^2 N.2 = a * b^-1 N.3 = b^-1 * a * b^2 * a^4 * b^-1 * a * b^2 N.4 = a * b^2 * a^3 * b^-1 * a^3 * b^3 > Order(N); 96 ``` The subgroup $H$ has a normaliser $N$ of order $96$. ```magma > Q := MaximalOvergroup(G, H); > Q; Finitely presented group Q on 3 generators Index in group G is 7 Generators as words in group G Q.1 = a^2 Q.2 = a * b^-1 Q.3 = b^-1 * a * b^2 > #Q; 1536 ``` So the largest subgroup $Q$ that contains $H$ has index $7$ in $G$. ```magma > M := NormalClosure(G, H); > M; Finitely presented group M on 7 generators Index in group G is 1 = 1 Generators as words in group G M.1 = a^2 M.2 = a * b^-1 M.3 = a^2 * b^-1 * a^-1 M.4 = b^-1 * a M.5 = b * a * b^-2 M.6 = (b^-1 * a * b)^2 M.7 = b^-1 * a ``` To summarise, proper subgroups of $G$ having orders 48, 96 and 1536 that contain $H$ have been found. The normal closure of $H$ is $G$. ## Properties of Subgroups The operations described in this subsection all require a closed coset table for at least one subgroup of an fp-group. If a closed coset table is needed and has not been computed, a coset enumeration will be invoked. If the coset enumeration does not produce a closed coset table, a runtime error is reported. Experienced users can control the behaviour of such indirectly invoked coset enumeration with a set of global parameters. These global parameters can be changed using the function [`SetGlobalTCParameters`](../FPGroups/subgroup-general.md#function-grpfp-setglobaltcparameters). For a detailed description of the available parameters and their meanings, refer to chapter [Finitely Presented Groups](../FPGroups/index-fpgroups.md#chapgrpfp). **Intrinsics:** `u in H` : True if and only if the word $u$ in $G$ is a member of $H$. `u notin H` : True if and only if the word $u$ in $G$ is not a member of $H$. `H eq K` : True if and only if subgroups $H$ and $K$ are equal. `H ne K` : True if and only if subgroups $H$ and $K$ are not equal. `H subset K` : True if and only if the subgroup $H$ is a subset of $K$. `IsConjugate`$(G, H, K)$ : True if and only if subgroups $H$ and $K$ are conjugate in $G$. If $H$ and $K$ are conjugate an element that conjugates $H$ into $K$ is also returned. `IsMaximal`$(G, H)$ : True if subgroup $H$ is maximal in $G$. `IsNormal`$(G, H)$ : True if subgroup $H$ is normal in $G$. `IsSelfNormalizing`$(G, H)$ : True if subgroup $H$ is a self-normalising subgroup of $G$. `IsPerfect`$(H)$ : True if the subgroup $H$ is perfect. ### `Example: Subgroup Preds (ex-b1118e)` Some of these subgroup predicates are illustrated by applying them to some subgroups of the two-dimensional space group $p4g = < r,\ s\ |\ r^2,\ s^4,\ (r, s)^2 >$. ```magma > p4g := FPGroup< r, s | r^2 = s^4 = (r,s)^2 = 1 >; > h := sub< p4g | (s^-1*r)^4, s*r >; > k := sub< p4g | (s^-1*r)^2, (s*r)^2 >; ``` Subgroups $h$ and $k$ have the same index in $p4g$ but they are neither equal nor conjugate. ```magma > Index(p4g, h), Index(p4g, k); 8 8 > h eq k; false > IsConjugate( p4g, h, k); false ``` Are subgroups $h$ and $k$ normal in $p4g$? ```magma > IsNormal(p4g, h), IsNormal(p4g, k); false true ``` Since $h$ is not normal it makes sense to construct its normal closure $n$ in $p4g$. It turns out that $n$ is maximal in $p4g$ and contains $k$. ```magma > n := NormalClosure(p4g, h); Index( p4g, n)l 2 > IsMaximal(p4g, n); true > k subset n; true ``` Here is a new subgroup $l$ of $p4g$ that turns out to be conjugate to $h$. ```magma > l := sub< p4g | (s*r)^4, s^-1*r >; > IsConjugate(p4g, h, l); true r^-1 ``` i.e. $l = h^{r^{-1}}$. The intersection of $h$ and $h^{r^{-1}}$ already yields the core of $h$ in $p4g$. ```magma > (h meet l) eq Core(p4g, h); true ``` ## Presentations for Subgroups Let $H$ be a subgroup of finite index in the finitely presented group $G$. It frequently happens that it is desirable to construct a set of defining relations for $H$ from those of $G$. Such a presentation can be obtained either on a set of Schreier generators for $H$ or on the given generators of $H$ using the Reidemeister–Schreier rewriting technique [[Magnus *et al.*, 1976](../../references.md#cite-magnus-cgt)], if necessary together with extended coset enumeration [[Arrell and Robertson, 1984](../../references.md#cite-extended-tc), [Havas *et al.*, 1984](../../references.md#cite-havas-tietze)]. It should be emphasised that if the user wishes only to determine the structure of the maximal abelian quotient of $H$, then the function `AbelianQuotientInvariants` should be used. In this case there is no need to first construct a presentation for $H$ using the `Rewrite` function described below, since `AbelianQuotientInvariants` employs a special form of the Reidemeister–Schreier rewriting process which abelianises each relator as soon as it is constructed. Thus, compared to the function `Rewrite`, the function `AbelianQuotientInvariants` can be applied to subgroups of much larger index. **Intrinsics:** `Rewrite` $(G,~H)$ : Given an fp group $G$ and a subgroup $H$ having finite index in $G$, return a group $K$ isomorphic to $H$ with a presentation on some of the Schreier generators of $H$ in $G$. The group $K$ is created as a subgroup of $G$ with a presentation on the new generators. The isomorphism from $H$ onto $K$ is returned as second return value. Note that the generators of $K$ will, in general, not correspond to the generators of $H$. `Rewrite` $(G,~\sim H)$ : Given a finitely presented group $G$ and a subgroup $H$ having finite index in $G$, a set of defining relations for $H$ is constructed on the given generators. The information stored for $H$ is changed to include these defining relations. **Notes:** **(i)** The first version of `Rewrite` selects its own generators and consequently is faster and often produces a simpler presentation. The second version constructs a presentation on the given generators. This is a more complicated process and the resulting presentation is often large. Consequently, by default, simplification is attempted using Tietze transformations that do not involve any changes to the generators of $H$. If is not necessary to obtain a presentation for $H$ on the original generators of $H$, the first method is strongly recommended. ### `Example: Rewrite (ex-d4d467)` Starting with the group $G$ defined by $$ < x, y\ |\ x^2,\ y^3,\ (xy)^{12},\ (xy)^6(xy^{-1})^6 >, $$ a subgroup $H$ of index $3$ is generated by the words $x$, $yxy^{-1}$ and $yxy^{-1}xy^{-1}xy$: ```magma > G := FPGroup< x, y | x^2, y^3, (x*y)^12, (x*y)^6*(x*y^-1)^6 >; > G; Finitely presented group G on 2 generators Relations x^2 = Id(G) y^3 = Id(G) (x * y)^12 = Id(G) x * y * x * y * x * y * x * y * x * y * x * y * x * y^-1 * x * y^-1 * x * y^-1 * x * y^-1 * x * y^-1 * x * y^-1 = Id(G) > H := sub< G | x, y*x*y^-1, y*x*y^-1*x*y^-1*x*y >; > H; Finitely presented group H on 3 generators Generators as words in group G H.1 = x H.2 = y * x * y^-1 H.3 = y * x * y^-1 * x * y^-1 * x * y > Index(G, H); 3 ``` A presentation for $H$ is now computed. Then using `AbelianQuotientInvariants`, $H$ is found to have a non-trivial soluble quotient. Using the $p$-quotient algorithm described later, $H$ is found to have a 2-quotient of order$2^{62}$! ```magma > K := Rewrite(G, H); > K; Finitely presented group K on 3 generators Generators as words in group G K.1 = x K.2 = y * x * y^-1 K.3 = x^y Relations K.1^2 = Id(T) K.2^2 = Id(T) K.3^2 = Id(T) (K.3 * K.2 * K.1 * K.3 * K.2)^2 = Id(K) (K.1 * K.3 * K.2 * K.1 * K.3)^2 = Id(K) (K.1 * K.2 * K.1 * K.3 * K.2)^2 = Id(K) > AbelianQuotientInvariants(K); [ 2, 2, 2 ] > Q2 := pQuotient(K, 2, 30); > FactoredOrder(Q2); [ <2, 62> ] ``` ### `Example: Rewrite2 (ex-282060)` This example demonstrates how the intrinsic `Rewrite` can be used to obtain a presentation of a finitely presented group on a different set of generators. Firstly, a presentation for $L_2(7)$ on two generators $x$ and $y$ is used to define the group. ```magma > L27 := FPGroup< x, y | x^3 = 1, y^3 = 1, (x*y)^4 = 1, > (y*y^x)^2 = y^x*y >; ``` The group is also generated by the elements $a = (xy)^2$ and $b = y$. ```magma > H := sub; > Index(L27, H); 1 ``` At this point, no defining relations of $H\cong L27$ on the generators $a$ and $b$ are known. ```magma > H; Finitely presented group H on 2 generators Index in group L27 is 1 Generators as words in group L27 a = (x * y)^2 b = y ``` The intrinsic `Rewrite` is applied to $H$ as a subgroup of $L27$ in order to compute defining relations on the generators $a$ and $b$. ```magma > Rewrite(L27, ~H); > H; Finitely presented group H on 2 generators Index in group L27 is 1 Generators as words in group L27 a = (x * y)^2 b = y Relations a^2 = Id(H) b^3 = Id(H) (a * b)^7 = Id(H) (a * b^-1 * a * b)^4 = Id(H) (b * a * b^-1 * a * b * a)^4 = Id(H) ``` The fifth relation turns out to be redundant: $a$ and $b$ are standard generators for $L_2(7)$ as is shown here: ```magma > Order(DeleteRelation(H, 5)) eq Order(H); true ```