All Questions
Tagged with lattice-orders group-theory
77
questions
5
votes
1
answer
123
views
A seeming example of a group whose subgroup lattice is lower semimodular but not consistent: where's my error?
Corollary 5.3.12 in Schmidt's "subgroup lattices of groups" states that if groups $A,B$ have lower semimodular subgroup lattices, then so does their direct product $A \times B$.
This paper ...
- 1,770
0
votes
0
answers
128
views
How to show that lattice of subgroups D4 isn't modular lattice?
Here is a lattice of subgroups D4.
The lattice isn't modular iff there is a "pentagon" as a sublattice.
As we can see $\left \{ \rho_{0} \right\} - \left \{ \rho_{0}, \mu_1 \right\} - \left ...
- 125
1
vote
1
answer
100
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Is the class of modular/distributive groups an axiomatizable class? [closed]
I define a modular group to be a group whose lattice of subgroups is a modular lattice, and similarly for distributive groups. My question is, are either of modular and/or distributive groups a first-...
- 21.4k
2
votes
0
answers
25
views
Reference request: finding maximal ordered subgroups of lattice-ordered groups?
I am trying to find references that provide techniques for constructing (totally-) ordered subgroups of lattice-ordered groups $G$. For those who are not familiar, a lattice-ordered group $G$ is set ...
3
votes
1
answer
153
views
Why is product of infimum and supremum of a commuting pair of elements in a lattice-ordered group equal to the product of the elements?
Context: Self-study.
Seth Warner's Modern Algebra (1965), question $15.11$ gives:
If $(G, \circ \preccurlyeq)$ is a lattice-ordered group and if $x$ and $y$ are commuting elements of $G$, then $$\sup ...
- 5,057
1
vote
1
answer
86
views
Semi-lattices whose Hasse Diagrams are trees after transitive reduction?
Is there a name or anything else known for a semi-lattice whose Hasse Diagram becomes a tree after applying transitive reduction?
Trying to find more about it since it comes up in an optimization ...
- 5,477
3
votes
1
answer
222
views
Help with proof that sublattice of normal subgroups of group $G$ is modular.
I am trying to prove that the set $N(G)$ of normal subgroups of a group $G$ is a sublattice of the lattice of subgroups of $G$, $Sub(G)$ where the meet operation ($\lor$) is defined as usual ...
- 311
2
votes
1
answer
56
views
cocyclic group and totally ordered lattice of subgroup
A group $G$ is said cocyclic if has a non-trivial minimum subgroup M or,equivantly, if the intersection M of all non-trivial subgroups of G is non-trivial.
For exemple the quaternion group $Q_8$ is ...
- 387
2
votes
0
answers
88
views
Can I make a lattice of the classes of subgroups related by outer automorphism for D8xC2?
While trying to come up with a coloring scheme to organize all the complex interactions of the D8×C2 lattice, I realized I can make a much simpler lattice like structure of the kind of subgroups ...
- 41
2
votes
4
answers
388
views
How's the 'integer lattice' or the 'direct sum' of $\mathbb{Z} \oplus \mathbb{Z}$ not a Free group?
My question is about the direct sum $\mathbb{Z} \oplus \mathbb{Z}$ which is a Free Abelian group and not a free group.
The the integer lattice, or what I think is the direct sum $\mathbb{Z} \oplus \...
- 69
2
votes
0
answers
51
views
If one subgroup of index $k$ is maximal normal in a join, are all subgroups of index $k$ maximal normal in this join?
I have been self-studying abstract algebra from Dummit & Foote for the past few months and recently learnt about the Jordan-Holder theorem. I was looking at this theorem from a subgroup lattice ...
- 143
0
votes
0
answers
51
views
comparison between partially ordered group and Semifield
I would like to compare partially ordered group $(G,\cdot,\leq)$ and semifield $(S,+,\cdot)$.
Here I want to find out the main differences between them. The similarities are clear for me. I think one ...
- 55
0
votes
0
answers
42
views
Can I combine lattices (group) and Minkowski's Theorem with crystallography?
I am studying computer science and have an interest in material science. sadly I am only a layman in mathematics and physics. but I want to understand those better to be able to use them more freely ...
- 1
0
votes
0
answers
69
views
Lattice structure of free abelian group on a set $X$
Let $X$ be a set and denote $\mathbb{Z}^{(X)}$ for the free abelian group on $X$. That is, $\mathbb{Z}^{(X)}$ consists of all the finite sums $\sum_{x \in X}n_xx$, with coefficients $n_x \in \mathbb{Z}...
user391447
3
votes
1
answer
626
views
Can two nonisomorphic groups have the same subgroup lattice?
Subgroup lattice is sometimes said to represent the structure of a group.But does it give a complete information about the group.I mean from a given subgroup lattice is it possible to construct a new ...
- 3,619