In mathematics, the lattice of subgroups of a group G {\displaystyle G} is the lattice whose elements are the subgroups of G {\displaystyle G} , with the...

case of subgroups of Rn, this amounts to the usual geometric notion of a lattice as a periodic subset of points, and both the algebraic structure of lattices...

divisibility lattice. In the finite case, the lattice of subgroups of a cyclic group of order n is isomorphic to the dual of the lattice of divisors of n, with...

lattice theorem, and variously and ambiguously the third and fourth isomorphism theorem) states that if N {\displaystyle N} is a normal subgroup of a...

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...

only if ST = TS. If S and T are subgroups of G, their product need not be a subgroup (for example, two distinct subgroups of order 2 in the symmetric group...

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. Lemma. Suppose G...

subgroups of a group is modular. But in general the lattice of all subgroups of a group is not modular. For an example, the lattice of subgroups of the...

with this lattice of translational symmetry cannot have more, but may have less symmetry than the lattice itself. A full list of subgroups is available...

degree 3 over Q {\displaystyle \mathbb {Q} } since the subgroups have index 3 in G. The subgroups are not normal in G, so the subfields are not Galois or...

if every maximal chain of subgroups has the same length. This is important to those interested in the lattice of subgroups of a group, and is sometimes...

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,...

mathematics, in the field of group theory, a modular subgroup is a subgroup that is a modular element in the lattice of subgroups, where the meet operation...

of all topological groups that are covered by the universal covering group form a lattice, corresponding to the lattice of subgroups of the center of...

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...

G is a maximal normal subgroup if and only if the quotient G/N is simple. These Hasse diagrams show the lattices of subgroups of the symmetric group S4...

to a degree the finite groups, with quasidihedral Sylow 2-subgroups. The Sylow 2-subgroups of the following groups are quasidihedral: PSL3(Fq) for q ≡...

modularity relationship. The definition encapsulates many of the nice properties of lattices of subgroups of supersolvable groups. A finite group G {\displaystyle...

associated with congruence identities. Quotient ring Congruence lattice problem Lattice of subgroups A. G. Kurosh, Lectures on General Algebra, Translated from...

ISBN 978-1-4822-4582-0 Suzuki, Michio (1951), "On the lattice of subgroups of finite groups", Transactions of the American Mathematical Society, 70 (2): 345–371...

Neumann conjecture. The lattice of subgroups of a group satisfies the ascending chain condition if and only if all subgroups of the group are finitely...

correspondence normal subgroups correspond to normal subgroups. This theorem is sometimes called the correspondence theorem, the lattice theorem, and the fourth...

The subgroups of any given group under inclusion. (While the infimum here is the usual set-theoretic intersection, the supremum of a set of subgroups is...

of these subgroups are distinct from each other, and apart from the trivial group {0} = 0Z, they all are isomorphic to Z. The lattice of subgroups of...

importance of the existence of normal subgroups. A subgroup N {\displaystyle N} of a group G {\displaystyle G} is called a normal subgroup of G {\displaystyle...

only if every pair of elements in the group generates a cyclic group. A group is locally cyclic if and only if its lattice of subgroups is distributive (Ore...

arrangement of points Lattice (discrete subgroup), a discrete subgroup of a topological group whose quotient carries an invariant finite Borel measure Lattice (module)...

with respect to the product of subgroups. The term quasinormal subgroup was introduced by Øystein Ore in 1937. Two subgroups are said to permute (or commute)...

modular group if its lattice of subgroups is modular. Alternatively, a group G is called an Iwasawa group when every subgroup of G is permutable in G...

Discrete normal subgroups play an important role in the theory of covering groups and locally isomorphic groups. A discrete normal subgroup of a connected...