Modular group (M group)

A modular group or M group is in the mathematical branch of group theory looked kind of groups . These are groups whose association of subgroups is a modular association .

Definitions

If and are subgroups of a group , then you place ${\displaystyle U}$ ${\displaystyle V}$ ${\displaystyle G}$

${\displaystyle U\land V:=U\cap V}$   =   Intersection of the subgroups

${\displaystyle U\lor V:=\langle U\cup V\rangle }$   = subgroup created by the union .

Through and the set of subgroups becomes an association with the trivial subgroups as the smallest and largest element. ${\displaystyle \land }$ ${\displaystyle \lor }$ ${\displaystyle {\mathcal {U}}}$

In general, this association is not modular, that is, the so-called modular law generally does not apply:

From     follows   . ${\displaystyle U\subset W}$ ${\displaystyle \forall V\in {\mathcal {U}}:\,U\lor (V\land W)=(U\lor V)\land W}$

A subgroup is called modular, if ${\displaystyle V\in {\mathcal {U}}}$

${\displaystyle \forall U\subset W\in {\mathcal {U}}:\,U\lor (V\land W)=(U\lor V)\land W}$.

A group is called a modular group or M-group if the subgroup association is modular, that is, if each subgroup is modular.

Hasse diagram of the subgroup association of D ₄ , the subgroups are represented by their cycle graphs .

D ₄ is not modular

The dihedral group D ₄ is not modular, as you can easily see from the adjacent representation of the subgroup lattice, because it obviously contains a subgroup that is isomorphic to N ₅ . The violation of the modular law can be specified in concrete terms: be the leftmost two-element subgroup, the four -element subgroup lying above it and the right-most two-element subgroup. : From to read , therefore, and ${\displaystyle U}$ ${\displaystyle W}$ ${\displaystyle V}$ ${\displaystyle V\land W=\{e\}}$ ${\displaystyle U\lor V=D_{4}}$

${\displaystyle U\lor (V\land W)=U\subsetneq W=(U\lor V)\land W}$.

So it's not modular, it's the smallest non-modular group. is an example of a non-modular subgroup. ${\displaystyle D_{4}}$ ${\displaystyle V\subset D_{4}}$

Comparison with Dedekind groups

Normal dividers are modular subgroups.

Therefore, Dedekind groups are modular, because by definition these are exactly the groups in which every subgroup is normal.

In particular, all Abelian groups are modular.

Among the Dedekind groups there are also non-Abelian groups; these are called Hamilton groups. The quaternion group Q ₈ is therefore an example of a non-Abelian modular group.

There are modular groups that are not Dedekind groups, see example below.

Finite p-groups

A finite p-group is modular if and only if every subquotient of the order is. For 2 groups, this means that non-modular groups must have a subquotient that is isomorphic to the dihedral group D ₄ , for p> 2 there must be a non-modular subquotient of the order in the non-modular case . ${\displaystyle p^{3}}$ ${\displaystyle p^{3}}$

The structure of the modular finite p-groups was discovered by K. Iwasawa in 1941 :

A p-group is modular if and only if holds ${\displaystyle G}$

(a) ( Hamiltonian , non-Abelian case) ${\displaystyle G\cong Q_{8}\times \mathbb {Z} _{2}^{n},\quad n\geq 0}$

or

(b) There is an Abelian normal divisor with a cyclic factor group as well as a and a natural number (with if ) such that from is generated and . ${\displaystyle N\subset G}$ ${\displaystyle G/N}$ ${\displaystyle x\in G}$ ${\displaystyle s}$ ${\displaystyle s\geq 2}$ ${\displaystyle p=2}$ ${\displaystyle G}$ ${\displaystyle N\cup \{x\}}$ ${\displaystyle xax^{-1}=a^{1+p^{s}}\,\forall a\in N}$

Examples

We clarify the above structure theorem of Iwasawa by groups of order 16.

A non-Hamiltonian, nonabelian group of order 16

We present here an example of a non-Hamiltonian and non-Abelian group with 16 elements. Note that the multiplication by 5 modulo 8 is an automorphism on the cyclic group . It is because this is the multiplication by 25 and that is the identity since 25 is equal to 1 modulo 8. thats why ${\displaystyle \alpha }$ ${\displaystyle \mathbb {Z} _{8}}$ ${\displaystyle \alpha \circ \alpha =\mathrm {id} _{\mathbb {Z} _{8}}}$

${\displaystyle \theta :\mathbb {Z} _{2}\rightarrow \mathrm {Aut} (\mathbb {Z} _{8}),\,\theta (0)=\mathrm {id} _{\mathbb {Z} _{8}},\,\theta (1)=\alpha }$

a homomorphism of into the automorphism group of and one can form the semi-direct product . ${\displaystyle \mathbb {Z} _{2}}$ ${\displaystyle \mathbb {Z} _{8}}$ ${\displaystyle G=\mathbb {Z} _{8}\rtimes _{\theta }\mathbb {Z} _{2}}$

The link in this group is

${\displaystyle (a,b)\cdot (c,d)=(a+5^{b}c;b+d),\quad a,c\in \{0,\ldots ,7\},\,b,d\in \{0,1\}}$,

where in the first component modulo 8 is calculated and in the second modulo 2.

In the above theorem by Iwasawa there is an Abelian normal divisor, is cyclic, for holds and for all ${\displaystyle N=\mathbb {Z} _{8}\times \{0\}}$ ${\displaystyle G/N\cong \mathbb {Z} _{2}}$ ${\displaystyle x=(0,1)\in G}$ ${\displaystyle G=\langle N\cup \{x\}\rangle }$ ${\displaystyle a\in \{0,\ldots ,7\}}$

${\displaystyle x\cdot (a,0)\cdot x=(0,1)\cdot (a,0)\cdot (0,1)=(0+5^{1}a,1)\cdot (0,1)=(5a,1)\cdot (0,1)=(5a+5^{1}\cdot 0,1+1)=(5a,0)=(a,0)^{5}=(a,0)^{1+2^{2}}}$

This group thus fulfills part (b) of Iwasawa's theorem above, and is therefore modular. It has elements of the following orders :

order	number	elements
1	1	the neutral element
2	3	(0.1), (4.0), (4.1)
4th	4th	(2, b), (6, b)
8th	8th	(a, b), a odd

All real subgroups are Abelian, because, according to Lagrange's theorem, only 8-element subgroups that are isomorphic to or to the dihedral group would come into question as non -Abelian subgroups (see list of small groups ). The former has 6 elements of order 4 and the latter has 5 elements of order 2, so they cannot be included in the above list of orders . In particular, the group is not Hamiltonian because otherwise it would have to contain a copy of . ${\displaystyle Q_{8}}$ ${\displaystyle D_{4}}$ ${\displaystyle G}$ ${\displaystyle Q_{8}}$

A non-modular group of order 16

In the above example we replace the automorphism with the multiplication by 3, which is also an automorphism on . Again , because this is the multiplication by 9 modulo 8 and therefore the identity. So we have homomorphism ${\displaystyle \alpha }$ ${\displaystyle \beta }$ ${\displaystyle \mathbb {Z} _{8}}$ ${\displaystyle \beta \circ \beta =\mathrm {id} _{\mathbb {Z} _{8}}}$

${\displaystyle \vartheta :\mathbb {Z} _{2}\rightarrow \mathrm {Aut} (\mathbb {Z} _{8}),\,\vartheta (0)=\mathrm {id} _{\mathbb {Z} _{8}},\,\vartheta (1)=\beta }$

and can thus form the semi-direct product . The link in this group is ${\displaystyle {\tilde {G}}=\mathbb {Z} _{8}\rtimes _{\vartheta }\mathbb {Z} _{2}}$

${\displaystyle (a,b)\cdot (c,d)=(a+3^{b}c;b+d),\quad a,c\in \{0,\ldots ,7\},\,b,d\in \{0,1\}}$,

where the first component is calculated again modulo 8 and in the second modulo 2.

Again is an Abelian normal divisor, is cyclic, holds for and for all ${\displaystyle N=\mathbb {Z} _{8}\times \{0\}}$ ${\displaystyle {\tilde {G}}/N\cong \mathbb {Z} _{2}}$ ${\displaystyle x=(0,1)\in {\tilde {G}}}$ ${\displaystyle {\tilde {G}}=\langle N\cup \{x\}\rangle }$ ${\displaystyle a\in \{0,\ldots ,7\}}$

${\displaystyle x\cdot (a,0)\cdot x=(0,1)\cdot (a,0)\cdot (0,1)=(0+3^{1}a,1)\cdot (0,1)=(3a,1)\cdot (0,1)=(3a+3^{1}\cdot 0,1+1)=(3a,0)=(a,0)^{3}}$.

This does not satisfy condition (b) from the above Iwasawa theorem for the case p = 2. In fact, this is the quasi-dihedral group with 16 elements, and it is not modular. With

${\displaystyle U:=\{(0,0),(0,1)\},\ V=\{(0,0),(2,1)\},\,W=\{(0,0),(0,1),(4,0),(4,1)\}}$

is easy to confirm

${\displaystyle U\lor (V\land W)=U\subsetneq W=(U\lor V)\land W}$.

Alternatively, one can use the above characterization by means of subquotients, taking into account that the quasi-dihedral group contains a subgroup that is isomorphic to the dihedral group , which is a non-modular subquotient of the order ^2-3 and therefore cannot be modular. ${\displaystyle D_{4}}$

Sub-modular subgroups

The modularity of subgroups is not a transitive property, that is, is a modular subgroup of the group and a modular subgroup of , then in general is not a modular subgroup of . Therefore the following concept of the submodular subgroup is introduced, which represents the transitive envelope of the relation " is modular subgroup in ": ${\displaystyle U}$ ${\displaystyle G}$ ${\displaystyle V}$ ${\displaystyle U}$ ${\displaystyle V}$ ${\displaystyle G}$

A subgroup is called submodular if there are subgroups ${\displaystyle U\subset G}$

${\displaystyle U=U_{0}\subset U_{1}\subset \ldots \subset U_{n}=G}$

there so that modular subset of is for each . ${\displaystyle U_{i}}$ ${\displaystyle U_{i+1}}$ ${\displaystyle i=0,\ldots ,n-1}$

See also

• Dedekind's modular law

Individual evidence

1. ^ Irene Zimmermann: Submodular subgroups in finite groups , Mathematische Zeitschrift (1989), Volume 202, 4, pages 545-557
2. ↑ Michio Suzuki: Structure of a Group and the Structure of its Lattice of Subgroups , Springer-Verlag 1956, ISBN 978-3-642-52760-9 , chap. I, §4: Finite Groups with a Modular Lattice of Subgroups
3. ↑ LN Shevrin, AJ Ovsyannikov: Semigroups and Their Subsemigroup Lattices , Springer-Verlag (2013), ISBN 9-401-58751-5 , Chapter II, 6.4
4. ^ Roland Schmidt: Subgroup Lattices of Groups , De Gruyter Mouton (1994), ISBN 978-3-11-011213-9 , Lemma 2.3.3
5. ↑ K. Iwasawa: About the finite groups and the associations of their subgroups , J. Univ. Tokyo (1941), pp. 171-199.
6. ↑ Michio Suzuki: Structure of a Group and the Structure of its Lattice of Subgroups , Springer-Verlag 1956, ISBN 978-3-642-52760-9 , chap. I, §4: Theorem 14
7. ^ Roland Schmidt: Subgroup Lattices of Groups , De Gruyter Mouton (1994), ISBN 978-3-11-011213-9 , Theorem 2.3.1
8. ^ Irene Zimmermann: Submodular subgroups in finite groups , Mathematische Zeitschrift (1989), Volume 202, 4, pages 545-557