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In mathematics, the lattice of subgroups of a group is the lattice whose elements are the subgroups of , with the partial ordering being set inclusion. In this lattice, the join of two subgroups is the subgroup generated by their union , and the meet of two subgroups is their intersection .
Let be a locally compact group and a discrete subgroup (this means that there exists a neighbourhood of the identity element of such that = {}).Then is called a lattice in if in addition there exists a Borel measure on the quotient space / which is finite (i.e. (/) < +) and -invariant (meaning that for any and any open subset / the equality () = is satisfied).
The intersection of subgroups A and B of G is again a subgroup of G. [5] For example, the intersection of the x-axis and y-axis in under addition is the trivial subgroup. More generally, the intersection of an arbitrary collection of subgroups of G is a subgroup of G.
The 24-dimensional Leech lattice has a fixed-point-free automorphism of order 3. Identifying this with a complex cube root of 1 makes the Leech lattice into a 12 dimensional lattice over the Eisenstein integers, called the complex Leech lattice. The automorphism group of the complex Leech lattice is the universal cover 6 · Suz of the Suzuki group.
More generally, there is a monotone Galois connection (,) between the lattice of subgroups of (not necessarily containing ) and the lattice of subgroups of /: the lower adjoint of a subgroup of is given by () = / and the upper adjoint of a subgroup / of / is a given by (/) =.
Hasse diagram of the Zassenhaus "butterfly" lemma – smaller subgroups are towards the top of the diagram. In mathematics, the butterfly lemma or Zassenhaus lemma, named after Hans Zassenhaus, is a technical result on the lattice of subgroups of a group or the lattice of submodules of a module, or more generally for any modular lattice. [1] Lemma.
G is a supersolvable group with elementary abelian Sylow subgroups (a special type of A-group), (Hall 1937, Theorem 1 and 2). Later, in (Zacher 1953), a group is said to be complemented if the lattice of subgroups is a complemented lattice, that is, if for every subgroup H there is a subgroup K such that H ∩ K = 1 and H, K is
A chief series of subgroups forms a chief chain in the lattice of subgroups. [3] The partition lattice of a finite set is supersolvable. A partition is left modular in this lattice if and only if it has at most one non-singleton part. [3] The noncrossing partition lattice is similarly supersolvable, [11] although it is not geometric. [12]
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