September 30, 2023

Mathematical finite p-groups with at most equation subgroups not in Chermak-Delgado lattice

Mathematical finite p-groups with at most $p^2+p$ subgroups not in Chermak-Delgado lattice

What groups are

In order for a set or system of elements to be called a mathematical group it needs to satisfy the following properties:

1.Closure

If there is a group G and elements $a,b\in G$ then $(a\cdot b)\in G$ .
Or
If there is a group G and elements $a,b\in G$ then $(a+b)\in G$ .

2.Associativity

If there is a group G and elements $a,b,c\in G$ then $(a\cdot b)\cdot c=a\cdot(b\cdot c)$ .

3.Inverse for every element

If there is a group G and element $a\in G$ and has inverse $a'\in G$ where $a'=\displaystyle\frac{1}{a}$ . Or
If there is a group G and element $a\in G$ and has inverse $a'\in G$ where $a'=-a$ .

4.Identity element

If there is a group G and element $a\in G$ and has inverse $a'\in G$ where $a'=\displaystyle\frac{1}{a}$ and $a\cdot a'=i$ where $i$ is the identity element.

What p-groups are

A group G is a p-group if the order of the group is a power of a prime i.e. p. Throughout evolving actions or operations will a group of order $|G|=p^n\cdot m$ where $p$ doesn't divide at $m$ , and $p^n$ is the highest power of dividing |G|.
P-group lemma: If the p-group G acts on a set A via $\theta:G\rightarrow Permutation(A)$ , then: $Fixed(\theta)\equiv_p|A|$ .
Normalizer lemma: If H is a p-subgroup of G and if the normalizer is $N_G(H)$ then, $[N_G(H):H]\equiv_p[G:H]$ .

Finite p-groups with at most subgroups not in Chermak-Delgado lattice

Lattices are an important subject and aspect in mathematics as well as useful for the applications lattice theory offer to physics applied mathematics and other sciences and other subjects. They help define order and through order, relations, chains and sets as well as other other things they are defined. The Chermak–Delgado lattice of G, denoted 𝒞𝒟(G), is the set of all subgroups with maximal Chermak–Delgado measure; this set is a modular sublattice within the subgroup lattice of G.
Suppose that G is a finite group, and H is a subgroup of G. The Chermak-Delgado measure of H (in G) is denoted by mG(H), and defined as mG(H) = |H| · |CG(H)|. The maximal Chermak-Delgado measure of G is denoted by m∗ (G), and defined as
m∗ (G) = max{mG(H) | H ≤ G}.
Let
. Then the set CD(G) forms a sublattice of L(G), the subgroup lattice of G, and is called the Chermak-Delgado lattice of G.
Following D. Burrell, W. Cocke and R. McCulloch [13], we use δCD(G) to denote the number of subgroups of G not in CD(G). That is,
δCD(G) = |L(G)| − |CD(G)|.
Our main result is stated as follows:
Let G be a finite p-group. Then δCD(G) 6 p 2 + p if and only if G is one of the following groups:
(1) Cp k , where k = 1, 2, . . . , $p^2$ + p; (In this case, δCD(G) = k.)
(2) Cp t × Cp, where t = 1, 2, . . . , p − 1; (In this case, δCD(G) = t(p + 1) + 1.)
(3) Q8; (In this case, δCD(G) = 1.)
(4) ha, b | a p k = b p = 1, ab = a 1+p k−1 i, where k = 2, 3, . . . , p + 1. (In this case, δCD(G) = (k − 1)(p + 1).)

Comments