A duality on the finite Abelian groups A, B is a bilinear mapping (, ) of \(A\times B\) into \({\mathbb{Q}}/{\mathbb{Z}}\), such that no non-zero element of A or B is orthogonal to the whole of the other group. Given A, there exist groups B for which a duality exists and there is a natural isomorphism between any two such B; further they are isomorphic to A. If P is a subgroup of A and \(P^{\perp}\) is the orthogonal subgroup of B, there is a duality on A/P and \(P^{\perp}\), so \(| P| | P^{\perp}| =| A|\) and \(P^{\perp \perp}=P\). In the case when \(B=A\), we say that P is regular if the restriction of (, ) to \(P\times P\) is a duality; in this case \(A=P\oplus P^{\perp}\). The Sylow subgroups of A are regular, and the Sylow p-subgroup of a maximal isotropic subgroup of A is maximal isotropic in the Sylow p-subgroup of A.
The present paper is concerned with symplectic modules G, which are Abelian groups on which a skew-symmetric duality is defined, and the aim is the study of maximal isotropic submodules H, which are called Lagrangian; we have \(| H|^ 2=| G|\). Also rk \(H\leq rk G\leq 2rk H\), and this cannot be improved. It is shown (3.3) that any two Lagrangians are extensions of their intersection by isomorphic groups. In 4.1, a symplectic module \(S_ 1(H)\) is constructed with given Lagrangian H, and it is shown that any symplectic module is isomorphic as such to \(S_ 1(H)\) for some Lagrangian H of it. It follows that if G is a symplectic module, rk G is even. Another description of the Lagrangians is given in 4.2. To deal with the general case we may suppose that G is an Abelian p-group, in view of the above remark. If \(e_ 1,e_ 1,...,e_ r,e_ r\) are the invariants of G and \(f_ 1,...,f_{2r}\) those of a Lagrangian K, both written in non-increasing order, then \(f_ 1+f_{2i}\geq e_ i\geq f_{2i-1}+f_{2r}\) for \(i=1,...,r\), and this yields a classification of Lagrangians of modules of rank 2 or 4. The symplectic module G is called almost homogeneous if its invariants are \(e_ 1,e_ 1,e,e,...,e,e\) where \(e_ 1\geq e\). The main result of §5 states that if K is a Lagrangian of such a module, then \(e_ 1\geq f_ k+f_{2r-k+1}\geq e\) for \(k=1,...,r\), where \(f_ 1,...,f_{2r}\) are the invariants of K. In §6, a lower bound for the order of an Abelian p- group is given which contains at least one Lagrangian of each symplectic module of order \(p^{2n}\). Generators and relations of G in terms of K are given in §7: G is generated by \(x_ 1,...,x_{2r}\) with the relations \(p^{f_ i}x_ i=0\) \((i=1,...,r)\), \(p^{f_ i}x_{r+i}=\sum s_{ij}x_ j\) \((i,j=1,...,r)\), where \((s_{ij})\) is a skew-symmetric matrix and \(| s_{ij}| \leq \min (p^{f_ i},p^{f_ j})\). This yields some necessary conditions for the invariants of Lagrangians of \(S_ 1(H)\). In §8 a second symplectic module \(S_ 2(H)\) with given Lagrangian H is constructed, and an application is given. The lower bound of §6 mentioned above is extended to regular (non-Abelian) p-groups in §9.