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.

2022年11月27日 · 定义 \text{(Lattice of Subgroups)} 一个群的子群格是由其所有子群和子群间的包含关系构成的 二元组 \left(\mathcal{S},\subseteq\right) 实际上就是子群所构成的 偏序集

2022年4月17日 · Given a group \(G\), the lattice of subgroups of \(G\) is the partially ordered set whose elements are the subgroups of \(G\) with the partial order relation being set inclusion. It is common to depict the subgroup lattice for a group using a Hasse diagram .

2017年1月30日 · Its subgroup lattice consists of $S_3$ at the top, the trivial group $\{e\}$ at the bottom, and four intermediate groups: $A_3$, which is generated by the permutation $(123)$, and the $2$-element subgroups generated by $(12)$, $(13)$, and $(23)$, respectively.

2021年9月25日 · Draw a subgroup lattice for \(\mathbb{Z}_{12}\text{.}\) The positive divisors of \(12\) are \(1,2,3,4,6,\) and \(12\text{;}\) so \(\mathbb{Z}_{12}\)'s subgroups are of the form \(\langle 1\rangle\text{,}\) \(\langle 2\rangle\text{,}\) etc.

In Lie theory and related areas of mathematics, a lattice in a locally compact group is a discrete subgroup with the property that the quotient space has finite invariant measure.

2021年9月25日 · It can be useful to look at how subgroups of a group relate to one another. One way of doing this is to consider subgroup lattices (also known as subgroup diagrams). To draw a subgroup lattice for a group \(G\text{,}\) we list all the subgroups of \(G\text{,}\) writing a subgroup \(K\) below a subgroup \(H\text{,}\) and connecting them with a ...

2014年12月19日 · Here's the subgroup lattice for the dihedral group of order 8 (labelled D4 in the diagram). As Timbuc mentioned, there are lots of (nicer!) diagrams of the subgroup lattice of D4 online. For other more obscure groups, you can use Sage (which uses GAP) to generate their subgroup lattice.

A lattice is a set with a partial order, such as the inclusion operator _< of subgroups, where each pair of members has a least upper bound and a greatest lower bound. For the pair {H, K} of subgroups the least upper bound is the subgroup <I-I, K>, and the greatest lower bound is the intersection H('~K.

With the two binary operations ∧ and ∨, L ⁢ (G) becomes a lattice. It is a bounded lattice, with G as the top element and e as the bottom element. Furthermore, if {H i ∣ i ∈ I} is a set of subgroups of G indexed by some set I, then both