Magma's Handbook devotes dozens of chapters to linear algebra and group theory. This card is a bird's-eye survey of two large clusters of that material: the module chapters (linear algebra over general rings) and the finite group chapters beyond permutation groups (matrix groups, polycyclic groups, black-box groups, almost-simple recognition, group databases, automorphisms, and cohomology). The goal here is orientation — knowing which chapter to reach for — not exhaustive coverage of any one of them. Permutation groups (GrpPerm) already have their own dedicated lessons and are only mentioned here for contrast.
Modules: the general linear-algebra layer
Every module in Magma is a submodule or quotient of a free module $S^{(n)}$ over some ring or algebra $S$. The Mod chapter is the conceptual entry point: "module" covers vector spaces, inner product spaces, $R$-modules for any ring $R$, group-ring modules $R[G]$, and the linear transformations between them. It also clarifies that rectangular matrices are themselves elements of $\Hom_R(M,N)$ — a matrix is a module homomorphism, viewed two ways at once.
ModRng is where the actual free-module machinery lives: tuple modules $R^{(n)}$ and matrix modules $\Hom_R(M,N)$, under the category hierarchy rooted at ModRng (with specializations like ModTupFld for tuples over a field). This is the workhorse category for basic module arithmetic, submodules, and quotients.
Two chapters specialize this picture. ModDed handles modules over Dedekind domains — torsion-free modules over the maximal order of a number field or function field, represented as pseudo-matrices (fractional ideals paired with coefficients, normalized to Hermite form) — the natural setting for lattices in algebraic number theory. ModCpx instead covers chain complexes: sequences of modules linked by boundary maps whose composition vanishes, with support for computing homology.
Finally, Multilinear generalizes modules to tensors: multilinear maps between sequences of modules, tensor spaces, categories of tensors, and their invariants. Where ModRng handles a single module or a map between two, Multilinear is for objects with several module "legs" at once.
> V := RSpace(Rationals(), 4); // tuple module over a field
> M := RModule(Integers(), 3); // free module over a general ring
Finite groups: the abstract layer above GrpPerm
The Grp chapter is the umbrella for everything group-theoretic that doesn't depend on how a group is represented. It enumerates the main categories Magma actually implements: GrpPerm (permutation groups, covered elsewhere), GrpMat (matrix groups, finite or infinite), GrpPC (finite soluble groups via power-conjugate presentation), GrpAb (abelian groups, finite or infinite), and GrpGPC (general polycyclic groups). Functions described in Grp itself — element construction, generic predicates — apply across whichever concrete category a group happens to use.
Matrix groups: finite fields, infinite fields, and Q/Z
Matrix groups form their own family of chapters because the right algorithm depends heavily on the base ring. GrpMatGen is the general introduction: a matrix group over a finite field or $\mathbb{Z}/m\mathbb{Z}$ with a short-enough orbit can get a BSGS (base and strong generating set) representation just like a permutation group. Over infinite rings like $\mathbb{Z}$, $\mathbb{Q}$, or a number field, Magma can sometimes decide finiteness but otherwise offers only basic arithmetic unless the group is recognized as a Lie-type group.
GrpMatFF picks up where BSGS becomes infeasible: matrix groups over finite fields too large or high-dimensional for a stabilizer chain. It layers on Aschbacher-category recognition (classifying maximal subgroups of $\mathrm{GL}(n,q)$) and the CompositionTree/LMG (large matrix group) framework of Monte Carlo and Las Vegas algorithms. GrpMatInf targets matrix groups over infinite fields directly, using congruence-homomorphism techniques to test finiteness and properties like soluble-by-finite or the Tits alternative. GrpMatQZ specializes further to finite matrix groups over $\mathbb{Q}$ and $\mathbb{Z}$: invariant forms, normalizers/centralizers inside $\mathrm{GL}_n(\mathbb{Z})$, and conjugacy, via lattice and perfect-form machinery.
Polycyclic and black-box representations
GrpPC covers finite soluble groups given by a power-conjugate presentation — generators $a_1,\dots,a_n$ with relations $a_j^{p_j} = w_{jj}$ and conjugation relations $a_j^{a_i} = w_{ij}$. This is the recommended representation whenever you need intensive computation with a soluble group, since the word problem and much of the structure theory become fully algorithmic.
GrpBB sits at the opposite extreme: a black-box group (GrpBB) is one where you can multiply, invert, and test equality of elements but nothing about the internal representation is exposed. It wraps an existing Magma group so algorithms designed for the black-box model — which underlie much recognition machinery elsewhere — can be applied.
Almost-simple recognition, databases, automorphisms, cohomology
Several chapters build on top of these representations to answer higher-level structural questions.
GrpASim provides tools for almost-simple groups: given a group $S$ known to be almost simple, either assert an isomorphism to a standard copy $T$ (non-constructive recognition) or build the isomorphism explicitly (constructive recognition), then transfer known facts about $T$ — conjugacy classes, maximal subgroups, Sylow subgroups — back to $S$.
GrpData is a library of precomputed group databases: all simple groups of order under $10^{20}$, all small groups up to order 2000 (plus several extended families), $p$-groups of order $p^n$ for $n \le 7$, metacyclic $p$-groups, and perfect groups up to order 50000 — useful for grabbing "the" group of a given order/type rather than constructing it from scratch.
GrpAuto computes automorphism groups: $U = \mathrm{Aut}(G)$, found via a lifting algorithm that climbs a characteristic series of $G$ layer by layer, using a database of automorphism groups of simple groups for the non-soluble composition factors.
GrpCoh provides cohomology and extension theory: computing $H^1(G,M)$ and $H^2(G,M)$ for a group $G$ acting on a module $M$, identifying cocycles, and constructing group extensions of $M$ by $G$.
Finally, GrpRed is a newer package for reductive groups — group schemes with trivial unipotent radical, currently focused on orthogonal and unitary groups — intended as infrastructure for representations of reductive groups and algebraic modular forms.
> A := AutomorphismGroup(G);
> CM := CohomologyModule(G, M);
> Dimension(CohomologyGroup(CM, 2));
Mental map
Mod/ModRng is generic linear algebra, ModDed specializes it to number-theoretic lattices, ModCpx and Multilinear generalize it to complexes and tensors. On the group side, Grp is the umbrella, the GrpMat* chapters split by base ring (finite field with/without BSGS, infinite field, $\mathbb{Q}/\mathbb{Z}$), GrpPC is for soluble groups and GrpBB for groups accessed only through a black box, and GrpASim, GrpData, GrpAuto, GrpCoh, GrpRed are specialized toolkits for simple-group recognition, databases, automorphisms, cohomology, and reductive groups.