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In mathematics, the lattice of subgroups of a group is the lattice ... "Generalized wreath products and the lattice of normal subgroups of a group" (PDF).
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 subgroups of any given group form a complete lattice under inclusion, called the lattice of subgroups. (While the infimum here is the usual set-theoretic intersection, the supremum of a set of subgroups is the subgroup generated by the set-theoretic union of the subgroups, not the set-theoretic union itself.)
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 (/) =.
Lattice of subfields (left) for / and inverted Lattice of subgroups of (/) The following is the simplest case where the Galois group is not abelian. Consider the splitting field K of the irreducible polynomial x 3 − 2 {\displaystyle x^{3}-2} over Q {\displaystyle \mathbb {Q} } ; that is, K = Q ( θ , ω ) {\displaystyle K=\mathbb {Q} (\theta ...
The lattice of parabolic subgroups of the dihedral group D 2×4, represented as a real reflection group, consists of the trivial subgroup, the four two-element subgroups generated by a single reflection, and the entire group. Ordered by inclusion, they give the same lattice as the lattice of fixed spaces ordered by reverse-inclusion.
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