**5. Solvable and supersolvable** *L***-subgroups of an** *L***-group**

The significance of the notion of solvability in classical group theory is beyond doubt. The studies pertaining to the notion of solvability and supersolvability frequently occur in the literature. The application of solvability in Galois Theory is established. Also, the application of nilpotency and solvability are well known in Lie groups.

#### **5.1 Solvable** *L***-subgroups of an** *L***-group**

In this subsection, we study the notion of derived series of an *L*-subgroup of an *L*-group in the same fashion as in classical group theory (see [29]). Here, also, *G* would denote a group that is not perfect.

Let *η*∈*L*ð Þ *μ* . We define inductively the following sequence of *L*-subgroups of *μ*:

$$
\eta^{(0)} = \eta \text{ and } \eta^{(i)} = \left[ \eta^{(i-1)}, \eta^{(i-1)} \right] \text{ for each } i. \tag{40}
$$

The following result is immediate:

Theorem 1.83 Let *<sup>η</sup>*∈*L*ð Þ *<sup>μ</sup>* . Then, *<sup>η</sup>*ð Þ*<sup>i</sup>* <sup>⊆</sup>*η*ð Þ *<sup>i</sup>*�<sup>1</sup> .

We study the concept of solvable *L*-subgroups of an *L*-group like its classical counterpart. For this purpose, we introduce the concept of a derived chain of an *L*-subgroup as follows:

Definition 1.17 Let *η*∈*L*ð Þ *μ* . Then, the chain

$$
\eta = \eta^{(0)} \supset \eta^{(1)} \supset \cdots \supset \eta^{(i)} \supset \cdots \tag{41}
$$

of *L*-subgroups of *η* is called the derived chain of *η*. Clearly, the tip of *η*ð Þ*<sup>i</sup>* coincides with *<sup>η</sup>*ð Þ*<sup>e</sup>* . Also, *<sup>η</sup>*ð Þ*<sup>i</sup>* <sup>∈</sup> NLð Þ*<sup>η</sup>* . Moreover, if *<sup>η</sup>*<sup>∈</sup> NLð Þ *<sup>μ</sup>* , then *<sup>η</sup>*ð Þ*<sup>i</sup>* <sup>∈</sup> NLð Þ *<sup>μ</sup>* .

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

Definition 1.18 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 derived chain

$$
\eta = \eta^{(0)} \supset \eta^{(1)} \supset \cdots \supset \eta^{(i)} \supset \cdots \tag{42}
$$

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 solvable *<sup>L</sup>*-subgroup of *<sup>μ</sup>*. If *<sup>m</sup>* is the least non negative integer such that *<sup>η</sup>*ð Þ *<sup>m</sup>* <sup>¼</sup> *<sup>η</sup><sup>a</sup>*<sup>0</sup> *<sup>t</sup>*<sup>0</sup> , then the series

$$
\eta = \eta^{(0)} \supseteq \eta^{(1)} \supseteq \cdots \supseteq \eta^{(m)} = \eta\_{t\_0}^{a\_0} \tag{43}
$$

is called the derived series of *η* and *m* is said to be the solvable length of *η*. If *η* is a solvable *L*-subgroup of *μ*, then we simply write *η* is solvable. Clearly, the tip and tail of the members *η*ð Þ*<sup>i</sup>* of derived 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'*η*ð Þ*<sup>i</sup>* ' of the derived chain which are peculiarities of *L*-setting.

Theorem 1.84 Let *η*∈*L*ð Þ *μ* and possesses sup-property. Then,

$$\operatorname{Im}\,\eta^{(i)}\subseteq\operatorname{Im}\,\eta\cup\{\operatorname{inf}^\*\,\eta\}.\tag{44}$$

Theorem 1.85 Let *η*∈*L*ð Þ *μ* and possesses sup-property. Then,

i. *η*ð Þ*<sup>i</sup>* possesses sup-property,

ii. *η*ð Þ*<sup>i</sup>* and *η* are jointly supstar.

The following result justifies the naturality of the extension of the notion of derived chain:

Theorem 1.86 Let *<sup>H</sup>* be a subgroup of *<sup>G</sup>*. Then for each *<sup>i</sup>*, 1ð Þ *<sup>H</sup>* ð Þ*<sup>i</sup>* <sup>¼</sup> <sup>1</sup>*H*ð Þ*<sup>i</sup>* .

For the justification of these notions, we also provide the level subset and strong level subset characterizations of the members of derived series.

Theorem 1.87 Let *η*∈*L*ð Þ *μ* and possesses sup-property. Then, for each *a*≰*inf η* and *<sup>a</sup>*≤*η*ð Þ*<sup>e</sup>* , *<sup>η</sup>*ð Þ*<sup>i</sup> <sup>a</sup>* <sup>¼</sup> *<sup>η</sup><sup>a</sup>* ð Þð Þ*<sup>i</sup>* for each *<sup>i</sup>*.

Theorem 1.88 Let *L* be a chain and *η*∈ *L*ð Þ *μ* . Then for each *a*, where inf *η*≤*a*<*η*ð Þ*e* , *η<sup>i</sup>* <sup>&</sup>gt; *<sup>a</sup>* <sup>¼</sup> *<sup>η</sup>*<sup>&</sup>gt; *a <sup>i</sup>* .

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

Theorem 1.89 Let *η*∈*L*ð Þ *μ* and possesses sup-property. Then, *η* is a solvable *L*subgroup of *μ* of solvable length at most n if and only if *η<sup>a</sup>* is a solvable subgroup of *μ<sup>a</sup>* of solvable length at most n for each *a*≰*inf η* and *a*≤*η*ð Þ*e* .

Theorem 1.90 Let *η*∈ *L*ð Þ *μ* and *L* be a chain. Then, *η* is a solvable *L*-subgroup of *μ* of solvable length at most *n* if and only if *η*<sup>&</sup>gt; *<sup>a</sup>* is a solvable subgroup of *μ*<sup>&</sup>gt; *<sup>a</sup>* of solvable length at most *n* for each *a*, where inf *η*≤*a*<*η*ð Þ*e* .

In view of the above results, the following is immediate:

Theorem 1.91 A subgroup *H* of a group of *G* is solvable if and only if 1*<sup>H</sup>* is a solvable *L*-subgroup of 1*G*.

Now, we provide the definition of a solvable series and use it to characterize solvable *L*-subgroups.

Definition 1.19 Let *<sup>η</sup>*∈*L*ð Þ *<sup>μ</sup>* be a proper *<sup>L</sup>*-subgroup with tip *<sup>a</sup>*<sup>0</sup> and tail *<sup>t</sup>*0. If *<sup>η</sup><sup>a</sup>*<sup>0</sup> *<sup>t</sup>*<sup>0</sup> is the trivial *L*-subgroup of *η*, then a series

$$
\eta = \eta\_0 \supseteq \eta\_1 \supseteq \cdots \supseteq \eta\_n = \eta\_{t\_0}^{a\_0} \tag{45}
$$

of *L*-subgroups of *η* is said to be a solvable series for *η*, if for each *i*

$$[\eta\_{i-1}, \eta\_{i-1}] \subseteq \eta\_i. \tag{46}$$

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

ð Þ*i η<sup>i</sup>*�1, *η<sup>i</sup>*�<sup>1</sup> ½ �⊆ *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<sup>i</sup>*�<sup>1</sup>ð Þ*y* for each *i*,

ð Þii *η<sup>i</sup>* ∈ *NL η<sup>i</sup>*�<sup>1</sup> ð Þ for each *i:*

The following is characterization relates the concept of solvability with that of solvable series:

Theorem 1.92 Let *η*∈ *L*ð Þ *μ* be a proper *L*-subgroup with tip *a*<sup>0</sup> and tail *t*0. Then, *η* is solvable if and only if *η* has a solvable series.

The above characterization helps us in obtaining the following:

Theorem 1.93 Let *η*, *θ* ∈*L*ð Þ *μ* be proper *L*-subgroups having identical tails. If *η* is a solvable *L*-subgroup and *θ* ⊆ *η*, then *θ* is also solvable.

Our next result establishes the fact that every nilpotent *L*-subgroup is solvable.

Theorem 1.94 Let *η*∈*L*ð Þ *μ* be a proper *L*-subgroup. Then, every central series of *η* is a solvable series.

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

In classical group theory, every nilpotent group is solvable. We obtain the same result in *L*-setting:

Theorem 1.95 Let *η*∈*L*ð Þ *μ* be a proper *L*-subgroup. If *η* is nilpotent, then *η* is also solvable.

The nature of image and pre-image of a solvable *L*-subgroup under a group homomorphism is ascertained in the following results:

Theorem 1.96 Let *f* : *G* ! *H* be a group homomorphism and *η*∈*L*ð Þ *μ* . Let *inf η* ¼ *inf f*ð Þ*η* . If *η* is solvable, then *f*ð Þ*η* is also solvable.

Theorem 1.97 Let *f* : *G* ! *H* be a group homomorphism having solvable kernel and *<sup>ν</sup>*∈*L H*ð Þ. Let *<sup>ρ</sup>*<sup>∈</sup> *<sup>L</sup>*ð Þ*<sup>ν</sup>* and *inf <sup>ρ</sup>* <sup>¼</sup> *inf f* �<sup>1</sup> ð Þ*ρ* . If *ρ* is solvable, then *f* �1 ð Þ*ρ* is also solvable.

So far in our studies, we have not dealt with normal series and subinvariant series in *L*-setting. Here we introduce these concepts and utilize them to characterize solvable *L*-subgroups (see [32]). We start with:

Definition 1.20 Let *η*∈*L*ð Þ *μ* be a proper *L*-subgroup with tip *a*<sup>0</sup> and tail *t*0. Then, a sequence *λ*0, *λ*1, … , *λ<sup>n</sup>* of *L*-subgroups of *μ* is said to be a normal (subinvariant) series of *η* if

$$
\eta = \lambda\_0 \supseteq \lambda\_1 \supseteq \dots \supseteq \lambda\_n = \eta\_{t\_0}^{a\_0} \tag{47}
$$

and *λ<sup>i</sup>* ∈ *NL*ð Þ *μ* ð Þ *λ<sup>i</sup>* ∈ *NL*ð Þ *λ<sup>i</sup>*�<sup>1</sup> for each *i*.

Remark: If *λ*0, *λ*1, … , *λ<sup>n</sup>* is a normal (subinvariant) series of *η*, then *λi*ð Þ¼ *e η*ð Þ¼ *e a*<sup>0</sup> and inf *λ<sup>i</sup>* ¼ inf *η* ¼ *t*<sup>0</sup> for each *i*.

Definition 1.21 Let *η*∈*L*ð Þ *μ* be a proper *L*-subgroup with tip *a*<sup>0</sup> and tail *t*0. Then, a normal (subinvariant) series *<sup>λ</sup>*<sup>0</sup> <sup>¼</sup> *<sup>η</sup>* <sup>⊇</sup> *<sup>λ</sup>*<sup>1</sup> <sup>⊇</sup> … <sup>⊇</sup> *<sup>λ</sup><sup>n</sup>* <sup>¼</sup> *<sup>η</sup><sup>a</sup>*<sup>0</sup> *<sup>t</sup>*<sup>0</sup> of *η* is said to be a normal (subinvariant) series with Abelian factors if for each *i*, the factor group ð Þ *<sup>λ</sup>i*�<sup>1</sup> *<sup>a</sup>* ð Þ *λ<sup>i</sup> <sup>a</sup>* is Abelian

for each *t*<sup>0</sup> <*a*≤*a*0.

The following theorem extends a famous result of classical group theory pertaining to the notion of solvability to *L*-setting:

Theorem 1.98 Let *L* be a dense chain and *η*∈ *L*ð Þ *μ* be a proper *L*-subgroup with tip *a*<sup>0</sup> and tail *t*0. Then, the following are equivalent:

i. *η* is solvable,

ii. *η* has a normal series with Abelian factors,

iii. *η* has a subinvariant series with Abelian factors.

Below we propose a definition of supersolvable subgroups in *L*-setting:

Definition 1.22 Let *η*∈*L*ð Þ *μ* be a proper *L*-subgroup with tip *a*<sup>0</sup> and tail *t*0. Then, *η* is said to be a supersolvable *L*-subgroup if *η* has a normal series *λ*<sup>0</sup> ¼

*<sup>η</sup>* <sup>⊇</sup> *<sup>λ</sup>*<sup>1</sup> <sup>⊇</sup> … <sup>⊇</sup> *<sup>λ</sup><sup>n</sup>* <sup>¼</sup> *<sup>η</sup><sup>a</sup>*<sup>0</sup> *<sup>t</sup>*<sup>0</sup> such that each factor group ð Þ *<sup>λ</sup>i*�<sup>1</sup> *<sup>a</sup>* ð Þ *λ<sup>i</sup> <sup>a</sup>* for each *a*, where *t*<sup>0</sup> < *a*≤ *a*0, is cyclic.

It has been established earlier, that *L*-subgroups of nilpotent *L*-subgroups and solvable *L*-subgroups are nilpotent and solvable respectively. It is well known that the class of supersolvable groups is also closed under the formation of subgroups in classical group theory. Here, we extend this property of supersovability to *L*-group theory.

Theorem 1.99 Let *η*∈*L*ð Þ *μ* be a proper *L*-subgroup with tip *a*<sup>0</sup> and tail *t*0. Let *θ* ∈*L*ð Þ*η* be such that *η* and *θ* have identical tips and also identical tails. If *η* is supersolvable, then *θ* is also supersolvable.

The nilpotency of the commutator of a supersolvable subgroup is also retained in *L*-setting. The result is as follows:

Theorem 1.100 Let *L* be an upper well ordered chain and *η*∈ *L*ð Þ *μ* be a proper *L*-subgroup with tip *a*<sup>0</sup> and tail *t*0. If *η* is supersolvable, then the commutator *L*-subgroup ½ � *η*, *η* is nilpotent.

#### **5.2 Zassenhaus theorem and Schreier refinement theorem**

In the earlier subsection, we have already introduced the concept of a normal (subinvariant) series of an *L*-subgroup of an *L*-group. In classical group theory, two normal series of a group are said to be equivalent if it is possible to set up a one to one correspondence between the factors of two series such that the paired factors are isomorphic. This is obtained by a certain type of factorization of a group into factor groups. Then, a generalization of second isomorphism theorem which is called Zassenhaus Theorem is used to establish this fact. Here in *L*-setting for the equivalence of two normal (subinvariant) series, we consider the factorization of each level of the *L*-subgroups in the spirit of classical group theory. Then, we extend Zassenhaus Theorem to *L*-setting and utilize it to establish a version of Schreier Refinement Theorem.

Below we extend the definitions of a refinement of a normal (subinvariant) series and equivalent normal (subinvariant) series (see [32]):

Definition 1.23 Let *η*∈ *L*ð Þ *μ* with tip *a*<sup>0</sup> and tail *t*<sup>0</sup> such that *a*<sup>0</sup> 6¼ *t*0. A normal (subinvariant) series of *η*

$$
\eta = \theta\_0 \supsetneq \theta\_1 \supsetneq \theta\_2 \supsetneq \dots \supsetneq \theta\_m = \eta\_{t\_0}^{a\_0} \tag{48}
$$

is said to be a refinement of a normal (subinvariant) series of *η*

$$
\eta = \eta\_0 \supseteq \eta\_1 \supseteq \eta\_2 \supseteq \dots \supseteq \eta\_n = \eta\_{t\_0}^{a\_0} \tag{49}
$$

if *η*0, *η*1, *η*2, … , *η<sup>n</sup>* is a subsequence of *θ*0, *θ*1, *θ*2, … , *θm*.

Definition 1.24 Let *η*∈*L*ð Þ *μ* with tip *a*<sup>0</sup> and tail *t*<sup>0</sup> such that *a*<sup>0</sup> 6¼ *t*0. Then, two normal (subinvariant) series *<sup>η</sup>* <sup>¼</sup> *<sup>η</sup>*<sup>0</sup> <sup>⊇</sup> *<sup>η</sup>*<sup>1</sup> <sup>⊇</sup> *<sup>η</sup>*<sup>2</sup> <sup>⊇</sup> … <sup>⊇</sup> *<sup>η</sup><sup>n</sup>* <sup>¼</sup> *<sup>η</sup><sup>a</sup>*<sup>0</sup> *<sup>t</sup>*<sup>0</sup> and *η* ¼

*<sup>θ</sup>*<sup>0</sup> <sup>⊇</sup> *<sup>θ</sup>*<sup>1</sup> <sup>⊇</sup> *<sup>θ</sup>*<sup>2</sup> <sup>⊇</sup> … <sup>⊇</sup> *<sup>θ</sup><sup>m</sup>* <sup>¼</sup> *<sup>η</sup><sup>a</sup>*<sup>0</sup> *<sup>t</sup>*<sup>0</sup> of an *L*-subgroup *η* are said to be equivalent if for each fixed *a*≤*a*0, the factors of the series

$$
\eta\_a = (\eta\_0)\_a \supseteq (\eta\_1)\_a \supseteq (\eta\_2)\_a \supseteq \dots \supseteq (\eta\_m)\_e = \{e\} \tag{50}
$$

can be put in one to one correspondence with the factors of the series

$$\eta\_a = (\theta\_0)\_a \supseteq (\theta\_1)\_a \supseteq (\theta\_2)\_a \supseteq \dots \supseteq (\theta\_m)\_e = \{e\} \tag{51}$$

in such a way that the paired factors are isomorphic.

Now, we extend Zassenhaus Theorem to *L*-setting:

Theorem 1.101 (Zassenhaus Theorem) Let *η*, *θ* ∈*L*ð Þ *μ :* Let *η*<sup>1</sup> ∈ *NL*ð Þ*η* and *θ*<sup>1</sup> ∈ *NL*ð Þ*θ* . Then,

$$(\eta \cap \theta\_1) \circ \eta\_1 \lhd (\eta \cap \theta) \bullet \eta\_1 \text{ and } (\eta\_1 \cap \theta) \bullet \theta\_1 \lhd (\eta \cap \theta) \bullet \theta\_1. \tag{52}$$

Also, there is an isomorphism such that

$$\frac{(\eta \cap \theta)\_a (\eta\_1)\_a}{(\eta \cap \theta\_1)\_a (\eta\_1)\_a} \cong \frac{(\eta \cap \theta)\_a (\theta\_1)\_a}{(\eta\_1 \cap \theta)\_a (\theta\_1)\_a} \text{ for each } a \le a\_0;\tag{53}$$

where *a*<sup>0</sup> ¼ *η*1ð Þ*e* ∧*θ*1ð Þ*e :*

The following theorem provides an extension of Schreier Refinement Theorem to *L*-setting:

Theorem 1.102 (Schreier Refinement Theorem) Let *L* be an upper well ordered chain. Then, every two normal series of an *L*-subgroup have refinements that are equivalent.
