**4. Commutator** *L***-subgroup and nilpotent** *L***-subgroup of an** *L***-group**

The class of nilpotent groups constitutes an important class in the studies of group theory. In fact, nilpotent groups are very near to Abelian groups and they are always solvable. They arise in the studies of Galois theory as well as in the classification of groups. In the investigation of nilpotent groups and solvable groups, the notion of commutator and commutator subgroups play very significant role. Therefore, we start with notion of commutator *L*-subgroups.

**Figure 2.** *Inter-relationship of the sublattices of L*ð Þ *μ .*

### **4.1 Commutator** *L***-subgroup of an** *L***-group**

In this subsection, we study the notion of a commutator in *L*-setting [28, 29] by using the notion of infimums of *L*-subsets. In fact, we also discuss here how infimums of *L*-subsets play an effective role in the development of the theory of *L*-subgroups.

Definition 1.13 Let *<sup>η</sup>*, *<sup>θ</sup>* <sup>∈</sup>*L<sup>μ</sup>*. Then, the commutator of *<sup>η</sup>* and *<sup>θ</sup>* is an *<sup>L</sup>*-subset ð Þ *<sup>η</sup>*, *<sup>θ</sup>* of *G* defined as follows:

$$(\eta, \theta)(x) = \begin{cases} \vee \{\eta(y) \wedge \theta(z)\}, & \text{if } x = [y, z] \text{ for some } y, z \in G, \\ \text{inf } \eta \wedge \text{inf } \theta, & \text{if } x \neq [y, z] \text{ for any } y, z \in G. \end{cases} \tag{28}$$

The commutator *L*-subgroup of *η*, *θ* ∈ *L<sup>μ</sup>* is defined as the *L*-subgroup of *G* generated by ð Þ *η*, *θ* . It is denoted by ½ � *η*, *θ* . Thus, ½ �¼ *η*, *θ* h i ð Þ *η*, *θ* . Moreover, ½ � *η*, *θ* ð Þ¼ *e η*ð Þ*e* ∧*θ*ð Þ*e* .

In general, if f g *η<sup>i</sup> n <sup>i</sup>*¼<sup>1</sup> <sup>⊆</sup> *<sup>L</sup><sup>μ</sup>*, then we write *<sup>η</sup>*1, *<sup>η</sup>*2, *<sup>η</sup>*3, … , *<sup>η</sup><sup>n</sup>* <sup>ð</sup> Þ ¼ *<sup>η</sup>*1, *<sup>η</sup>*<sup>2</sup> ð Þ, *<sup>η</sup>*<sup>3</sup> ð Þ, … , *<sup>η</sup><sup>n</sup>* ð Þ and *η*1, *η*2, *η*3, … , *η<sup>n</sup>* ½ � ¼ *η*1, *η*<sup>2</sup> ½ �, *η*<sup>3</sup> ½ �, … , *η<sup>n</sup>* ½ �. Moreover, it follows that inf *η*1, *η*2, *η*3, … , *η<sup>n</sup>* ð Þ ¼ inf *η*1∧ inf *η*2∧ … ∧ inf *ηn*. Also, *η*1, *η*2, *η*3, … *η<sup>n</sup>* ½ �ð Þ¼ *e η*1ð Þ*e* ∧*η*2ð Þ*e* ∧ … ∧*ηn*ð Þ*e* .

*Development of* L*-Group Theory DOI: http://dx.doi.org/10.5772/intechopen.110387*

In the following, we present some extensions of the results of classical group theory with certain deviations and peculiarities:

Theorem 1.58 Let *<sup>η</sup>*, *<sup>θ</sup>* <sup>∈</sup>*L<sup>μ</sup>* and *<sup>η</sup>*<sup>⊆</sup> *<sup>θ</sup>*. Then, ½ � *<sup>η</sup>*, *<sup>σ</sup>* <sup>⊆</sup>½ � *<sup>θ</sup>*, *<sup>σ</sup>* for each *<sup>σ</sup>* <sup>∈</sup> *<sup>L</sup><sup>μ</sup>*. Theorem 1.59 Let *<sup>η</sup>*, *<sup>θ</sup>* <sup>∈</sup>*L<sup>μ</sup>*. Then, ½ �¼ *<sup>η</sup>*, *<sup>θ</sup>* ½ � *<sup>θ</sup>*, *<sup>η</sup>* . Theorem 1.60 Let *η*, *θ* ∈*L<sup>μ</sup>*. Then,

$$
\inf \eta \wedge \inf \theta \le \inf \eta \circ \theta \le \inf \eta \vee \inf \theta.\tag{29}
$$

Theorem 1.61 Let *η*, *θ* ∈ *NL*ð Þ *μ* and *σ* ∈*L*ð Þ *μ :* If either *η* and *θ* or *θ* and *σ* have the same tails, then

$$[\sigma \circ \eta, \theta] \subseteq [\eta, \theta] \circ [\sigma, \theta],\tag{30}$$

and moreover, if *η*ð Þ¼ *e σ*ð Þ*e* , then the equality holds. Theorem 1.62 Let *η*, *θ* ∈ *NL*ð Þ *μ* . Then,

$$[\eta, \theta] \in \mathcal{NL}(\mu) \text{ and } [\eta, \theta] \subseteq \eta \cap \theta. \tag{31}$$

Theorem 1.63 Let *<sup>η</sup>*, *<sup>θ</sup>* <sup>∈</sup> *<sup>L</sup><sup>μ</sup>* and f : *<sup>G</sup>* ! *<sup>K</sup>* be a group homomorphism. Then, fð Þ¼ h i*η* h i fð Þ*η* and fð Þ¼ ½ � *η*, *θ* h i fð Þ ð Þ *η*, *θ :*

Theorem 1.64 Let *<sup>η</sup>*, *<sup>θ</sup>* <sup>∈</sup>*CL<sup>μ</sup>* be jointly supstar. Then, ½ � *<sup>η</sup>*, *<sup>θ</sup>* <sup>∈</sup>*CLs*ð Þ *<sup>μ</sup>* .

Theorem 1.65 Let *<sup>η</sup>*, *<sup>θ</sup>* <sup>∈</sup> *<sup>L</sup><sup>μ</sup>* and f : *<sup>G</sup>* ! *<sup>K</sup>* be a homomorphism. Let inf *<sup>η</sup>* <sup>¼</sup> inf fð Þ*<sup>η</sup>* and inf *θ* ¼ inf fð Þ*θ* . Then, f½ �¼ ð Þ*η* , fð Þ*θ* fð Þ ½ � *η*, *θ* .

Theorem 1.66 Let f : *<sup>G</sup>* ! *<sup>H</sup>* be a homomorphism and *<sup>ν</sup>*∈*L H*ð Þ. Let *<sup>λ</sup>*, *<sup>σ</sup>* <sup>∈</sup>*L<sup>ν</sup>* and inf *<sup>λ</sup>* <sup>¼</sup> inf f�<sup>1</sup> ð Þ*<sup>λ</sup>* , inf *<sup>σ</sup>* <sup>¼</sup> inf f�<sup>1</sup> ð Þ *<sup>σ</sup>* . Then, f�<sup>1</sup> ð Þ*<sup>λ</sup>* , f�<sup>1</sup> ð Þ *<sup>σ</sup>* <sup>⊆</sup> <sup>f</sup> �1 ð Þ ½ � *λ*, *σ* .

Theorem 1.67 Let *η*, *θ* ∈*L*ð Þ *μ* be such that *η* and *θ* are jointly supstar. Then, the commutator ð Þ *η*, *θ* and hence the commutator *L*-subgroup ½ � *η*, *θ* possess sup-property.

Note that, *a*≤ inf *η* if and only if *η<sup>a</sup>* ¼ *G*. Moreover, if *a*≤ inf *η*∧ inf *θ*, then the level subsets *ηa*, *θ<sup>a</sup>* and ð Þ *η*, *θ <sup>a</sup>* coincide with *G* and hence ð Þ *η*, *θ <sup>a</sup>* 6¼ *η<sup>a</sup>* ð Þ , *θ<sup>a</sup>* .

Theorem 1.68 Let *η*, *θ* ∈*L*ð Þ *μ* . If *a*<sup>0</sup> ¼ *η*ð Þ*e* ∧*θ*ð Þ*e* and *a*≰inf *η*∧ inf *θ*, then

$$\text{i. } [\eta, \theta]\_{\mathfrak{a}} = [\eta\_a, \theta\_a] \text{ for each } a \le a\_0 \text{, provided } \eta \text{ and } \theta \text{ are jointly support.}$$

ii. ½ � *<sup>η</sup>*, *<sup>θ</sup>* <sup>&</sup>gt; *<sup>a</sup>* <sup>¼</sup> *<sup>η</sup>*<sup>&</sup>gt; *<sup>a</sup>* , *θ* <sup>&</sup>gt; *a* for each *a*<*a*0, provided *L* is a chain.

In general, let f g *η<sup>i</sup> n <sup>i</sup>*¼<sup>1</sup> <sup>⊆</sup> *<sup>L</sup>*ð Þ *<sup>μ</sup>* . If *<sup>a</sup>*<sup>0</sup> <sup>¼</sup> *<sup>η</sup>*1ð Þ*<sup>e</sup>* <sup>∧</sup>*η*2ð Þ*<sup>e</sup>* <sup>∧</sup> … <sup>∧</sup>*ηn*ð Þ*<sup>e</sup>* and *<sup>a</sup>*≰inf *<sup>η</sup><sup>i</sup>* for each *<sup>i</sup>*, then


#### **4.2 Nilpotent** *L***-subgroup of an** *L***-group**

We begin this subsection with the concept of a descending central chain of an *L*-subgroup by making use of the notion of commutators. Then, this in turn has been used to define nilpotent *L*-subgroups [28]. Throughout this subsection, *G* would denote a group which is not perfect.

We start with the definition of a descending central chain of an *L*-subgroup *η* of an *L*-subgroup *μ*.

Take *Z*0ð Þ¼ *η η*, *Z*1ð Þ¼ *η* ½ � *Z*0ð Þ*η* , *η* . And in general, for each *i*, define

$$Z\_i(\eta) = [Z\_{i-1}(\eta), \eta]. \tag{32}$$

The following result is an immediate consequence of the above definition: Theorem 1.69 Let *η*∈ *L*ð Þ *μ* . Then for each *i*, *Zi*ð Þ*η* ⊆*Zi*�<sup>1</sup>ð Þ*η* . Here we provide the definition of a descending central chain. Definition 1.14 Let *η*∈*L*ð Þ *μ* . Then, the chain

*Development of* L*-Group Theory DOI: http://dx.doi.org/10.5772/intechopen.110387*

$$\eta = Z\_0(\eta) \supset Z\_1(\eta) \supset \cdots \supset Z\_i(\eta) \supset \cdots \tag{33}$$

of *L*-subgroups of *μ* is called the descending central chain of *η*.

It is worthwhile to note that as *η*∈ *NL*ð Þ*η* , in view of Theorem 1.62, *Zi*ð Þ*η* is a normal *L*-subgroup of *η* for each *i*. Moreover, if *η*∈*L*ð Þ *μ* , then *Zi*ð Þ*η* ∈ *NL*ð Þ *μ* .

Now, we are in a position to formulate the definition of a nilpotent *L*-subgroup of an *L*-group.

Definition 1.15 Let *η*∈*L*ð Þ *μ* with tip *a*<sup>0</sup> and tail *t*<sup>0</sup> and *a*<sup>0</sup> 6¼ *t*0. If the descending central chain

$$\eta = Z\_0(\eta) \supset Z\_1(\eta) \supset \cdots \supset Z\_i(\eta) \supset \cdots \tag{34}$$

terminates finitely to the trivial *L*-subgroup *η<sup>a</sup>*<sup>0</sup> *<sup>t</sup>*<sup>0</sup> , then *η* is known as a nilpotent *L*-subgroup of *μ*. More precisely, *η* is said to be nilpotent of class *c* if *c* is the least non-negative integer such that *Zc*ð Þ¼ *<sup>η</sup> <sup>η</sup><sup>a</sup>*<sup>0</sup> *<sup>t</sup>*<sup>0</sup> . In this case, the series

$$\eta = Z\_0(\eta) \supseteq Z\_1(\eta) \supseteq \cdots \supseteq Z\_{\varepsilon}(\eta) = \eta\_{t\_0}^{a\_0} \tag{35}$$

is called the descending central series of *η*. If it is a nilpotent *L*-subgroup of *μ*, then we simply write *η* is nilpotent. Clearly, the tip and tail of the members *Zi*ð Þ*η* of descending central series coincide with the tip and the tail of the trivial *L*-subgroup *η<sup>a</sup>*<sup>0</sup>

*t*0 . Next, we provide some results pertaining to the member '*Zi*ð Þ*η* ' of the descending central chain which are peculiarities of *L*-setting.

Theorem 1.70 Let *η*∈ *L*ð Þ *μ* and possesses sup-property. Then for each *i*,

$$ImZ\_i(\eta) \subseteq Im\eta \cup \{\inf \eta\}.\tag{36}$$

Theorem 1.71 Let *η*∈*L*ð Þ *μ* and possesses sup-property. Then for each *i*,

i. *Zi*ð Þ*η* possesses sup-property.

ii. *Zi*ð Þ*η* and *η* are jointly supstar.

The following result justifies the naturality of the extension of the notion of descending central chain:

Theorem 1.72 Let *H* be a subgroup of *G*. Then for each *i*, *Zi*ð Þ¼ 1*<sup>H</sup>* 1*Zi*ð Þ *<sup>H</sup>* .

For further justification of these notions, we provide the level subset and strong level subset characterizations of the members of descending central series.

Theorem 1.73 Let *η*∈*L*ð Þ *μ* and possesses sup-property. Then for each *a*≰inf *η* and *a*≤*η*ð Þ*e* , *Zi η<sup>a</sup>* ð Þ¼ ð Þ *Zi*ð Þ*η <sup>a</sup>* for each *i*.

Theorem 1.74 Let *L* be a chain and *η*∈*L*ð Þ *μ* . Then for each *a*, where inf *η*≤*a*<*η*ð Þ*e* , *Zi η*<sup>&</sup>gt; *a* <sup>¼</sup> ð Þ *Zi*ð Þ*<sup>η</sup>* <sup>&</sup>gt; *<sup>a</sup>* for each *i*.

Finally, we obtain the level subset and strong level subset characterizations for nilpotent *L*-subgroups.

Theorem 1.75 Let *η*∈*L*ð Þ *μ* and possesses sup-property. Then, *η* is a nilpotent *L*-subgroup of *μ* of nilpotent length at most *n* if and only if *η<sup>a</sup>* is a nilpotent subgroup of *μ<sup>a</sup>* of nilpotent length at most *n* for each *a* �≤ inf *η* and *a*≤*η*ð Þ*e* .

Theorem 1.76 Let *η*∈ *L*ð Þ *μ* and *L* be a chain. Then, *η* is a nilpotent *L*-subgroup of *μ* of nilpotent length at most *n* if and only if *η*<sup>&</sup>gt; *<sup>a</sup>* is a nilpotent subgroup of *μ*<sup>&</sup>gt; *<sup>a</sup>* of nilpotent length at most *n*, for each *a*, where inf *η*≤*a*<*η*ð Þ*e* .

The following result immediately follows from the above results:

Theorem 1.77 A subgroup *H* of a group of *G* is nilpotent if and only if 1*<sup>H</sup>* is a nilpotent *L*-subgroup of 1*G*.

Next, we provide the definition of a central chain and use it to characterize nilpotent *L*-subgroups.

Definition 1.16 Let *η*∈*L*ð Þ *μ* with tip *a*<sup>0</sup> and tail *t*0. Then, the chain

$$
\eta = \eta\_0 \supseteq \eta\_1 \supseteq \dots \supseteq \eta\_n \supseteq \dots \tag{37}
$$

of *L*-subgroups of *μ* is called a central chain of *η*, if for each *i*, *ηi*�<sup>1</sup> ½ � , *η* ⊆*ηi:* If *a*<sup>0</sup> 6¼ *t*<sup>0</sup> and there exists a positive integer *<sup>m</sup>* such that *<sup>η</sup><sup>m</sup>* <sup>¼</sup> *<sup>η</sup>a*<sup>0</sup> *<sup>t</sup>*<sup>0</sup> , where *ηa*<sup>0</sup> *<sup>t</sup>*<sup>0</sup> is the trivial *L*subgroup of *η* with tip *a*<sup>0</sup> and tail *t*0, then

$$
\eta = \eta\_0 \supseteq \eta\_1 \supseteq \dots \supseteq \eta\_m = \eta\_{t\_0}^{a\_0} \tag{38}
$$

is known as a central series of *η*. It follows that *η<sup>i</sup>* and *η* have identical tips and also, identical tails for each *i*. Moreover, the following is easy to verify:

i. *η<sup>i</sup>*�<sup>1</sup> ½ � , *η* ⊆ *η<sup>i</sup>* if and only if *ηi*ð Þ ½ � *x*, *y* ≥*η<sup>i</sup>*�<sup>1</sup>ð Þ *x* ∧*η*ð Þ*y* for each *i*,

ii. *η<sup>i</sup>* ∈ *NL*ð Þ*η* for each *i:*

Now, we are in a position to study the notion of a nilpotent *L*-subgroup by making use of the concept of a central chain. The results are as follows:

Theorem 1.78 Let *η*∈*L*ð Þ *μ* be a proper *L*-subgroup of *μ*. Then, *η* is nilpotent if and only if *η* has a central series.

Theorem 1.79 Let *η*∈ *L*ð Þ *μ* and *θ* be a proper *L*-subgroup of *η* such that *η* and *θ* have the common tail *t*0. If *η* is nilpotent, then *θ* is also nilpotent.

The notion of set product is an extension of the notion of product of complexes in classical group theory. The following two results provides a necessary mechanism for the set product of two nilpotent *L*-subgroups of *μ* to be nilpotent:

Theorem 1.80 Let *η*, *η*1, … , *η<sup>n</sup>*þ<sup>1</sup> ∈ *NL*ð Þ *μ* having identical tails.

If *η<sup>i</sup>* ¼ *η* for k + 1 distinct values of *i* where 0 ≤*k*≤*n*, then *η*1, *η*2, … , *η<sup>n</sup>*þ<sup>1</sup> <sup>⊆</sup>*Zk*ð Þ*<sup>η</sup> :* Theorem 1.81 Let *η* and *θ* be trivial *L*-subgroups of *μ*. Then, the set product *η* ∘ *θ* is also a trivial *L*-subgroup of *μ* defined by

$$\eta \circ \theta(\mathbf{x}) = \begin{cases} \eta(e) \wedge \theta(e) & \text{if } \,\, \mathbf{x} = e, \\ \inf \eta \vee \inf \theta & \text{if } \,\, \mathbf{x} \neq e, \end{cases} \tag{39}$$

if and only if *inf η*∨*inf θ* <*η*ð Þ*e* ∧*θ*ð Þ*e* .

The final result of this subsection provides a sufficient condition for the set product of two nilpotent *L*-subgroups to be nilpotent.

Theorem 1.82 Let *η*, *θ* ∈ NLð Þ *μ* with common tails *t*0, such that *t*<sup>0</sup> <*η*ð Þ*e* ∧*θ*ð Þ*e* and inf *η* ∘ *θ* ¼ *t*0. If *η* and *θ* are nilpotent of classes *c* and *d* respectively, then *η* ∘ *θ* is a nilpotent *L*-subgroup of *μ* of class at most *c* þ *d*.
