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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).
A lattice is modular if and only if it does not have a sublattice isomorphic to N 5 (shown in Pic. 11). [7] Besides distributive lattices, examples of modular lattices are the lattice of submodules of a module (hence modular), the lattice of two-sided ideals of a ring, and the lattice of normal subgroups of a group.
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
In mathematics, a supersolvable lattice is a graded lattice that has a maximal chain of elements, each of which obeys a certain modularity relationship. The definition encapsulates many of the nice properties of lattices of subgroups of supersolvable groups .
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.
These are always finite-index subgroups and the congruence subgroup problem roughly asks whether all subgroups are obtained in this way. The conjecture (usually attributed to Jean-Pierre Serre) is that this is true for (irreducible) arithmetic lattices in higher-rank groups and false in rank-one groups. It is still open in this generality but ...
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 ...