The L3-U3-quotient algorithm computes for a finitely presented group on two generators all quotients isomorphic to PSL(3,q), PGL(3, q), PSU(3, q), or PGU(3, q), simultaneously for all prime powers q.

A Magma implementation is available here: l3.tgz.
This includes an
algorithm to compute minimal associated primes.

The algorithm requires a Magma version >= 2.19.

To use the algorithm, after starting magma, type

> Attach("minass.magma"); > AttachSpec("l3.spec");

Note that the algorithm does **not** return images
onto the group PSL(3,2) = PSL(2,7).

- Basic usage
- Intermediate usage
- Advanced Usage
- Handling infinite L3-quotients
- Verbose printing
- References

The main method is L3Quotients, which takes a finitely presented group and computes all L3-quotients.

> G := Group< a,b | a^2, b^3, (a*b)^11, (a,b)^11 >; > L3Quotients(G); [ U_2(43) ] > H := Group< a,b | a^2, b^3, (a*b)^11, (a,b)^28 >; > L3Quotients(H); [ U_3(43), U_3(14057) ]This means that G has PSU(3,43) as quotient, but no other PSL(3,q), PGL(3, q), PSU(3, q), or PGU(3, q) is a quotient of G. Similarly, the only L3- or U3-quotients of H are PSU(3, 43) and PSU(3, 14057).

> G := Group< a,b | a^2, b^3, (a*b)^18, (a,b)^16 >; > L3Quotients(G); [ PGU(3, 71), U_3(1889), PGU(3, 17), PGU(3, 17), PGU(3, 17), PGL(3, 19) ]In this example, PGU(3,17) occurs three times. This means that there are three epimorphisms of G onto PGU(3,17) which do

Some groups have infinitely many L3-quotients. This is indicated by one of the L3-quotients L_3(infty^k), L_3(p^(infty^d)), or L_3(infty^(infty^d)). See below for an interpretation of this output, and how to use Magma to get more information about the quotients.

> G := Group< a, b | a^2, b^3, (a*b)^9 >; > L3Quotients(G); [ L_3(infty^18) ] > H := Group< a, b | a^2, b^3, (a,b)^5, (a, b*a*b*a*b)^3 >; > L3Quotients(H); [ L_3(2^infty) ] > K := Group< a, b | a^2, b^3 >; > L3Quotients(K); [ L_3(infty^(infty^2)) ]Even without further interpretation of the output, this tells us that these groups are all infinite.

> G := Group< a, b | a^2, b^3, (a*b)^11, (a,b)^28 >; > quot := L3Quotients(G); quot; [ U_3(43), U_3(14057) ] > H := GetMatrices(quot[2]); > H; MatrixGroup(3, GF(14057^2)) Generators: [ 13479*$.1 + 8387 13479*$.1 + 8388 12480*$.1 + 1163] [ 13727*$.1 + 1669 13727*$.1 + 1668 9677*$.1 + 9154] [ 11763*$.1 + 3700 11763*$.1 + 3700 908*$.1 + 4001] [ 562*$.1 + 11659 11987*$.1 + 12203 4961*$.1 + 13767] [11922*$.1 + 10773 5054*$.1 + 6130 3695*$.1 + 3079] [ 3814*$.1 + 8470 4617*$.1 + 9807 8441*$.1 + 10325]Note that G -> H, G.i -> H.i does in general not define a homomorphism, but the induced map G -> H/Z(H) does.

The orders can be specified using the optional parameter exactOrders. This is a list of pairs, where the first entry is a word in the group, and the second entry is the order.

There are two basic methods to further investigate such quotients. The first is SpecifyCharacteristic, which takes an L3-quotient and an integer n, and computes the L3-quotients in characteristic p|n.

> G := Group< a, b | a^2, b^3, (a*b)^9 >; > quot := L3Quotients(G); quot; [ L_3(infty^18) ] > Q := quot[1]; > SpecifyCharacteristic(Q, 103); [ PGL(3, 103^3) ] > SpecifyCharacteristic(Q, 107); [ U_3(107), U_3(107), U_3(107) ] > SpecifyCharacteristic(Q, 109); [ L_3(109), L_3(109), L_3(109) ] > SpecifyCharacteristic(Q, 113); [ PGU(3, 113^3) ]

> G< a, b > := Group< a, b | a^2, b^3, (a*b)^11 >; > quot := L3Quotients(G); quot; [ L_3(infty^20) ] > Q := quot[1]; > AddGroupRelations(Q, [(a,b)^32]); [ L_3(353^2), U_3(241) ]

Again, we can use AddGroupRelations to sudy this quotient further.

> H< a, b > := Group< a, b | a^2, b^3, (a,b)^5, (a, b*a*b*a*b)^3 >; > quot := L3Quotients(H); quot; [ L_3(2^infty) ] > Q := quot[1]; > AddGroupRelations(Q, [(a*b)^(2*3*5*7)]); [ PGL(3, 2^2), U_3(2^2) ]

> H< a, b > := Group< a, b | a^2, b^3, (a,b)^5, (a, b*a*b*a*b)^3 >; > quot := L3Quotients(H); quot; [ L_3(2^infty) ] > Q := quot[1]; > Q`Ideal; Ideal of Graded Polynomial ring of rank 10 over Integer Ring Order: Grevlex with weights [8, 2, 2, 2, 2, 4, 4, 4, 4, 1] Variables: xcom, x1, xm1, x2, xm2, x12, xm12, xm21, xm2m1, zeta Variable weights: [8, 2, 2, 2, 2, 4, 4, 4, 4, 1] Inhomogeneous, Dimension >0 Groebner basis: [ xcom + zeta + 1, xm12*xm2m1 + zeta, x12 + xm12, xm21 + xm2m1, x1 + 1, xm1 + 1, x2, xm2, zeta^2 + zeta + 1, 2 ] > R< xcom, x1, xm1, x2, xm2, x12, xm12, xm21, xm2m1, zeta > := Generic(Q`Ideal); > AddRingRelations(Q, [x12^5 + xm21^2 + 1]); [ L_3(2^8), L_3(2^6) ]

[1] |
S. Jambor An L3-U3-quotient algorithm for finitely presented groups PhD Thesis (2012). |