**6. Normal closure and subnormal** *L***-subgroups of an** *L***-group**

The study of a normal closure is important in classical group theory. The concept arises due to the fact that certain subgroups of a group are away from being normal. It is the the smallest subgroup containing a given subgroup which is normal in the group. This notion leads to some refined concepts in classical group theory such as normal closure series and subnormality.

#### **6.1 Normal closure series and subnormality**

In order to define a suitable notion of normal closure in *L*-setting which can be used to introduce the notion of subnormality, we start with the following (see [30]):

Definition 1.25 Let *<sup>η</sup>*∈*L*ð Þ *<sup>μ</sup>* . Define an *<sup>L</sup>*-subset *μημ*�<sup>1</sup> of *<sup>G</sup>* as follows:

$$\mu\eta\mu^{-1}(\mathbf{x}) = \bigvee\_{\substack{\boldsymbol{x} = \boldsymbol{x}\boldsymbol{\mu}^{-1} \\ \boldsymbol{y}, \boldsymbol{z} \in G}} \{\eta(\boldsymbol{y}) \wedge \mu(\boldsymbol{z})\} \quad \text{for each } \boldsymbol{x} \in G. \tag{54}$$

We call the *L*-subset *μημ*�1, conjugate of *η* in *μ*. Clearly, *η*⊆*μημ*�<sup>1</sup> ⊆*μ*. Moreover, ∨ *<sup>x</sup>*<sup>∈</sup> *<sup>G</sup> μημ*�<sup>1</sup>ð Þ *<sup>x</sup>* <sup>¼</sup> *<sup>η</sup>*ð Þ*<sup>e</sup>* and *μημ*�<sup>1</sup>ð Þ¼ *<sup>x</sup> μημ*�<sup>1</sup> *<sup>x</sup>*�<sup>1</sup> ð Þ for each *<sup>x</sup>*<sup>∈</sup> *<sup>G</sup>*. The normal closure of *η* in *μ* is defined as the *L*-subgroup of *μ* generated by the conjugate *μημ*�1. It is denoted by *ημ*. Thus *ημ* <sup>¼</sup> *μημ*�<sup>1</sup> .

The above defined notion satisfies the characteristic properties of a normal closure and retains the usual group theoretic relationship with the concepts of commutator subgroups and set product in *L*-setting. The result are as follows:

Theorem 1.103 Let *<sup>η</sup>*∈*L*ð Þ *<sup>μ</sup>* . Then, *ημ* is the least normal L-subgroup of *<sup>μ</sup>* containing *η*.

Theorem 1.104 Let *η*∈ *L*ð Þ *μ* . Then, ½ � *μ*, *η* ∘ *η*∈*L*ð Þ *μ* . Theorem 1.105 Let *η*, *θ* ∈ *L*ð Þ *μ* . Then,


Before we embark on the study of normal closure series and subnormality in *L*setting, we generalize notion of conjugacy and provide the definition of a conjugate of an *L*-subset by an *L*-subset (see [10, 33]).

Definition 1.26 Let *η*, *θ* ∈*L<sup>μ</sup>*. Define an *L*-subset *θηθ*�<sup>1</sup> of *G* as follows:

$$
\theta \eta \theta^{-1}(\varkappa) = \bigvee\_{\varkappa = \imath yx^{-1}} \{\eta(y) \wedge \theta(z)\} \quad \text{for each } \varkappa \in G. \tag{55}
$$

We call the *L*-subset *θηθ*�<sup>1</sup> the conjugate of *η* by *θ*. Clearly, *θηθ*�<sup>1</sup> ⊆ *μ:* Hence the *<sup>L</sup>*-subgroup *θηθ*�<sup>1</sup> <sup>∈</sup>*L*ð Þ *<sup>μ</sup>* and is denoted by *<sup>η</sup><sup>θ</sup>*.

Following theorem is instrumental in the development of this subsection: Theorem 1.106 Let *η*, *θ* ∈*L*ð Þ *μ* . Then,

$$\begin{aligned} \text{i. } &\eta \subseteq \theta \eta \theta^{-1} \quad \text{provided } \eta(e) \lessgtr \theta(e), \\\\ \text{ii. } &\bigvee\_{x \in G} \left\{ \theta \eta \theta^{-1}(x) \right\} = \theta \eta \theta^{-1}(e) = \eta(e) \wedge \theta(e), \\\\ \text{iii. } &\theta \eta \theta^{-1}(x) = \theta \eta \theta^{-1}(x^{-1}) \text{ for each } x \in G, \\\\ \text{iv. } &\theta \eta \theta^{-1}(\text{gxg}^{-1}) \ge \theta \eta \theta^{-1}(x) \wedge \theta(g) \text{ for each } x, g \in G. \end{aligned}$$

Firstly, we discuss here some properties of conjugate of *L*-subsets where the *L*subsets in question are *L*-subgroups. The significance of such properties have already been shown in classical group theory for establishing certain properties of *i*th normal closure of a subgroup of a group.

Theorem 1.107 Let *θ* ⊆ *γ* and *η*ð Þ¼ *e θ*ð Þ*e* . Then, Let *η*, *θ*, *γ* ∈ *L*ð Þ *μ* .

$$\left(\eta^{r}\right)^{\theta} = \eta^{r},\tag{56}$$

$$(\eta^{\theta})^{\check{\gamma}} = \eta^{\check{\gamma}},\tag{57}$$

$$
\eta^{\theta \circ \eta} = \eta^{\theta}.\tag{58}
$$

Theorem 1.08 Let *<sup>η</sup>*, *<sup>θ</sup>*, *<sup>γ</sup>* <sup>∈</sup>*L*ð Þ *<sup>μ</sup>* be such that *<sup>γ</sup>*ð Þ¼ *<sup>e</sup> <sup>η</sup>*ð Þ¼ *<sup>e</sup> <sup>θ</sup>*ð Þ*<sup>e</sup> :* Then, *ηθ <sup>γ</sup>* <sup>¼</sup> *ηγ* <sup>∘</sup> *<sup>θ</sup>*. Theorem 1.09 Let *<sup>η</sup>*, *<sup>θ</sup>* <sup>∈</sup>*L*ð Þ *<sup>μ</sup>* and *ηθ* <sup>¼</sup> *<sup>η</sup>:* Then, *<sup>η</sup>* <sup>∘</sup> *<sup>θ</sup>* <sup>∈</sup>*L*ð Þ *<sup>μ</sup>* .

In order to introduce the notion of subnormality of an *L*-subgroup of an *L*-group, we define a descending series. For *η*∈*L*ð Þ *μ* , define a series of *L*-subgroups of *μ* inductively as follows:

$$
\eta\_0 = \mu, \ \eta\_1 = \eta^{\mu}, \ \eta\_2 = \eta^{\eta\_1}, \ \dots, \eta\_i = \eta^{\eta\_{i-1}} \dots \tag{59}
$$

By Theorem 1.103, *η*<sup>1</sup> is the smallest normal *L*-subgroup of *μ* containing *η* and *η*<sup>2</sup> is the smallest normal *L*-subgroup of *η*<sup>1</sup> containing *η* and so on.Thus, we have

$$
\eta \sqsubseteq \dots \lhd \eta\_{i+1} \lhd \eta\_i \lhd \dots \lhd \eta\_1 \lhd \eta\_0 = \mu. \tag{60}
$$

This inductively defined series is known as the normal closure series of *η* in *μ* and we call *η<sup>i</sup>* the *i*th normal closure of *η* in *μ*. It is easy to verify that *ηi*ð Þ¼ *e η*ð Þ*e* for each *i*.

Theorem 1.110 Let *η*, *θ* ∈ *L*ð Þ *μ* such that *η*ð Þ¼ *e θ*ð Þ*e :* Let *η<sup>i</sup>* be the ith normal closure of *η* in *μ* and *ηθ <sup>i</sup>* ¼ *ηi:* Then, *η<sup>i</sup>*þ<sup>1</sup> ∈ *NL η<sup>i</sup>* ð Þ ∘ *θ :*

Now, define the notion of a subnormal *L*-subgroup of an *L*-group as follows:

Definition 1.27 Let *η*∈*L*ð Þ *μ* and *η<sup>i</sup>* be the *ith* normal closure of *η* in *μ*. If there exists a non negative integer *m* such that

$$
\eta\_m = \eta \lhd \eta\_{m-1} \lhd \cdots \lhd \eta\_0 = \mu,\tag{61}
$$

then *η* is known as a subnormal *L*-subgroup of *μ* with defect *m*. We shall denote a subnormal *L*-subgroup *η* of *μ* with defect *m* by *η* ⊲ *m μ*. If *η* is a subnormal *L*-subgroup of *μ*, then we shall write *η* is subnormal in *μ*.

Remark: Obviously *m* equals 0 if *η* ¼ *μ* and *m* ¼ 1 if *η*∈ *NL*ð Þ *μ* and *η* 6¼ *μ*.

The following theorem shows that the normal closure series is the fastest descending normal series [33]:

Theorem 1.111 Let *η*∈ *L*ð Þ *μ* and

$$
\eta \subseteq \cdots \lhd \eta\_{i+1} \lhd \eta\_i \lhd \cdots \lhd \eta\_1 \lhd \eta\_0 = \mu \tag{62}
$$

be the normal closure series of *η*. If there exists a descending series *γ*<sup>0</sup> ¼ *μ*, *γ*1, … , *γ<sup>i</sup>* … of *L*-subgroups of *μ* such that

$$
\eta \subseteq \dots \lhd \gamma\_{i+1} \lhd \dots \lhd \gamma\_0 = \mu,\tag{63}
$$

then *η<sup>i</sup>* ⊆ *γi:*

Following is the definition of a subnormal series of an *L*-group:

Definition 1.28 Let *η*∈*L*ð Þ *μ* . A finite series *θ*<sup>0</sup> ¼ *μ*, *θ*1, *θ*2, … , *θ<sup>m</sup>* ¼ *η* of *L*-subgroups of *μ* such that

$$
\eta = \theta\_m \lhd \theta\_{m-1} \lhd \dots \lhd \theta\_0 = \mu \tag{64}
$$

is said to be a subnormal series of *η:*.

We shall describe the notion of a subnormal *L*-subgroup through the notion of above defined subnormal series. The following result inter-connects these two concepts:

Theorem 1.112 Let *η*∈*L*ð Þ *μ :* Then, *η* is a subnormal *L*-subgroup of *μ* having defect *m* if and only if *η* has a subnormal series

$$
\eta = \gamma\_m \qplus \dots \lhd \gamma\_{i+1} \qplus \dots \gamma\_1 \lhd \gamma\_0 = \mu,\tag{65}
$$

of length *m* and *m* is the smallest length of such a subnormal series. The results given below are established with the help of the above theorem: Theorem 1.113 Let *η* be a subnormal *L*-subgroup of *μ* with defect *m*.


It can be seen easily that the intersection of any finite set of subnormal *L*-subgroups is again subnormal. More generally:

Theorem 1.114 Let *θ<sup>i</sup>* f g : *i* ∈*I* be a family of subnormal *L*-subgroups such that defect of *θ<sup>i</sup>* is *mi* where *mi* ≤ *m:* Then, ∩ *i*∈*I θ<sup>i</sup>* is a subnormal *L*-subgroup of *μ* with defect *c* where *c*≤ *m*.

Our next result determines the transitivity of the notion of subnormality.

Theorem 1.115 Let *η*, *θ* ∈*L*ð Þ *μ* such that *η* is a subnormal *L*-subgroup of *θ* with defect *m* and *θ* is a subnormal *L*-subgroup of *μ* with defect *n*. Then, *η* is a subnormal *L*-subgroup of *μ* with defect *m* þ *n*.

The following theorem establishes that the subnormality in *L*-setting is also preserved under the action of a homomorphism and its inverse image:

Theorem 1.116 Let *η*∈*L*ð Þ *μ* and *f* : *G* ! *K* be a group homomorphism. Then,


#### **6.2 Subnormal** *L***-subgroups and nilpotency**

In this subsection, we characterize subnormal *L*-subgroups by the usual group theoretic subnormality of the level subsets of the given *L*-subgroups. We shall refer this as a level subset characterization of subnormality. Then, this characterization is used to establish that when the lattice *L* is an upper well ordered chain, then every *L*-subgroup of a nilpotent *L*-group is subnormal. For this purpose, we need to develop a necessary mechanism (see [10, 33]). Here, we present:

Theorem 1.117 Let *η*∈ *L*ð Þ *μ* be such that *μ* and *η* are jointly supstar. Then,

$$\operatorname{Im}\,\eta^{\mu}\subseteq\operatorname{Im}\,\left(\mu\eta\mu^{-1}\right)\subseteq\operatorname{Im}\mu\cup\operatorname{Im}\eta.\tag{66}$$

More generally we have:

Theorem 1.118 Let *η*∈ *L*ð Þ *μ* be such that *μ* and *η* are jointly supstar. Then, for each i

$$\operatorname{Im}\,\eta\_{i+1}\subseteq\operatorname{Im}\,\left(\eta\_{i}\eta\eta\_{i}^{-1}\right)\subseteq\operatorname{Im}\mu\cup\operatorname{Im}\eta,\tag{67}$$

where *η<sup>i</sup>* is the *ith* normal closure of *η* in *μ*.

Theorem 1.119 Let *η*∈*L*ð Þ *μ* be such that *μ* and *η* are jointly supstar. Then for each *i*,


Below, we discuss the level subsets and strong level subsets of the normal closure of *η* in *μ*.

Theorem 1.120 Let *η*∈*L*ð Þ *μ :* Then,

i. *<sup>η</sup><sup>μ</sup>* ð Þ*<sup>a</sup>* <sup>¼</sup> *<sup>η</sup><sup>a</sup>* ð Þ*<sup>μ</sup><sup>a</sup>* for each *<sup>a</sup>*<sup>≤</sup> *<sup>η</sup>*ð Þ*<sup>e</sup>* , provided *<sup>μ</sup>* and *<sup>η</sup>* are supstar,

$$\text{iii.} \left(\eta^{\mu}\right)\_{a}^{>} = \left(\eta\_{a}^{>}\right)^{\mu\_{a}^{>}} \text{ for each } a < \eta(e), \text{ provided} \\ L \text{ is a chain.} $$

Theorem 1.121 Let *η*∈*L*ð Þ *μ* be such that *η* and *μ* are jointly supstar. Then, *η* is subnormal having defect at most *n* if and only if each level subset *η<sup>a</sup>* is subnormal having defect atmost *n* where *a*≤*η*ð Þ*e* .

Theorem 1.112 Let *G* be a group and *H* be its subgroup. Then, *H* is a subnormal subgroup of *G* if and only if 1*<sup>H</sup>* is a subnormal *L*-subgroup of 1*G*.

The strong level subset characterization of subnormal *L*-subgroup can be obtained easily.

Theorem 1.123 Let *L* be chain and *η*∈*L*ð Þ *μ* . Then, *η* is subnormal having defect at most *n* if and only if each strong level subset *η* <sup>&</sup>gt; *<sup>a</sup>* is subnormal having defect atmost *n* where *a*<*η*ð Þ*e* .

Theorem 1.124 Let *L* be an upper well ordered chain. Let *η*∈*L*ð Þ *μ* and *η* be nilpotent having tip *a*<sup>0</sup> and tail *t*<sup>0</sup> and *a*<sup>0</sup> 6¼ *t*0. If *θ* ∈*L*ð Þ*η* and has the tail *t*0, then *θ* is a subnormal *L*-subgroup of *η*.

In a forthcoming paper [10], we develop a mechanism in order to tackle the join problem for subnormal *L*-subgroup and we prove that:

Theorem 1.125 Let *η* and *θ* be subnormal *L*-subgroups of *μ*. Let *η*ð Þ¼ *e θ*ð Þ*e* and *η* ∘ *θ* ∈ *L*ð Þ *μ* . Then, the following are equivalent:

