# Trying to get the point of some Group Theory Lemmas

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• jstrunk
In summary: So ##x_1^{−1}x_2x_3^{−2}x_2^{-1} \in F(X)##, but ##x_1^{−1}x_2x_3^{−2}x_2^{-1}x_1 \notin F(X)##, but ##x_1^{−1}x_2x_3^{−2}x_2^{-1}x_1 \in G##. So ##F(X) \subseteq G##. However, we could also have ##X=\{\,x_1\,\}## and then ##F(X)=\{\,x_1^{n}\,|\,n \in \
jstrunk
TL;DR Summary
There are two related Lemmas in Schaum's Outline of Group Theory, Chapter 4 that seem excessively convoluted. Either I am missing something or they can be made much simpler and clearer.
There are two related Lemmas in Schaum's Outline of Group Theory, Chapter 4 that seem excessively convoluted.
Either I am missing something or they can be made much simpler and clearer.

Lemma 4.2:
If H is a subgroup of G and ${\rm{X}} \subseteq {\rm{H}}$ then ${\rm{H}} \supseteq \left\{ {{\rm{x}}_1^{{\varepsilon _1}}...{\rm{x}}_n^{{\varepsilon _n}}|{x_i} \in X,{\varepsilon _i} = \pm 1,n{\text{ a positive integer}}} \right\}$.

Lemma 4.3
Let G be a group and let X be a non-empty subset of G.
Let ${\rm{S = }}\left\{ {{\rm{x}}_1^{{\varepsilon _1}}...{\rm{x}}_n^{{\varepsilon _n}}|{x_i} \in X,{\varepsilon _i} = \pm 1,n{\text{ a positive integer}}} \right\}$. Then S is a subgroup of G. If H is any subgroup containing X, then ${\rm{H}} \supseteq {\rm{S}}$.

Question1: In Lemma 4.2, does H being a subgroup of G add anything? I think we could replace "If H is a subgroup of G" with "If H is group".
Question2: In Lemma 4.3, the isn't statement
If H is any subgroup containing X, then ${\rm{H}} \supseteq {\rm{S}}$ exactly equivalent to Lemma 4.2?

So the two Lemmas seem to amount to this (if we replace that expression in curly braces with gp(X)):
If H is a group and ${\rm{X}} \subseteq {\rm{H}}$ then ${\rm{gp}}\left( X \right) \subseteq H$.
If H is a group and $\left( {{\rm{X}} \ne \emptyset } \right) \subseteq {\rm{H}}$ then ${\rm{gp}}\left( {\rm{X}} \right)$ is a subgroup of H.

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jstrunk said:
Summary: There are two related Lemmas in Schaum's Outline of Group Theory, Chapter 4 that seem excessively convoluted. Either I am missing something or they can be made much simpler and clearer.

There are two related Lemmas in Schaum's Outline of Group Theory, Chapter 4 that seem excessively convoluted.
Either I am missing something or they can be made much simpler and clearer.

Lemma 4.2:
If H is a subgroup of G and ${\rm{X}} \subseteq {\rm{H}}$ then ${\rm{H}} \supseteq \left\{ {{\rm{x}}_1^{{\varepsilon _1}}...{\rm{x}}_n^{{\varepsilon _n}}|{x_i} \in X,{\varepsilon _i} = \pm 1,n{\text{ a positive integer}}} \right\}$.

Lemma 4.3
Let G be a group and let X be a non-empty subset of G.
Let ${\rm{S = }}\left\{ {{\rm{x}}_1^{{\varepsilon _1}}...{\rm{x}}_n^{{\varepsilon _n}}|{x_i} \in X,{\varepsilon _i} = \pm 1,n{\text{ a positive integer}}} \right\}$. Then S is a subgroup of G. If H is any subgroup containing X, then ${\rm{H}} \supseteq {\rm{S}}$.

Question1: In Lemma 4.2, does H being a subgroup of G add anything? I think we could replace "If H is a subgroup of G" with "If H is group".
Why should we restrict the statement? With respect of Lemma 4.3. it might be useful that it holds for any subgroup, ##G=H## included.
Question2: In Lemma 4.3, the isn't statement
If H is any subgroup containing X, then ${\rm{H}} \supseteq {\rm{S}}$ exactly equivalent to Lemma 4.2?
Lemma 4.3. says that ##S## is the smallest possible group which contains ##X##. Lemma 4.2. does not contain the statement, that the words over ##X## are actually a group, which is the first part. Only the second part is the same as Lemma 4.2.
So the two Lemmas seem to amount to this (if we replace that expression in curly braces with gp(X)):
If H is a group and ${\rm{X}} \subseteq {\rm{H}}$ then ${\rm{gp}}\left( X \right) \subseteq H$.
If H is a group and $\left( {{\rm{X}} \ne \emptyset } \right) \subseteq {\rm{H}}$ then ${\rm{gp}}\left( {\rm{X}} \right)$ is a subgroup of H.
This is not quite correct. We need two sets ##F(X)=\left\{ {{\rm{x}}_1^{{\varepsilon _1}}...{\rm{x}}_n^{{\varepsilon _n}}|{x_i} \in X,{\varepsilon _i} = \pm 1,n{\text{ a positive integer}}} \right\}## and ##gp(X)=\bigcap \{\,H \leq G\,|\,X \subseteq H \,\}##. ##F(X)## is a set, the set of all words over ##X##. ##gp(X)## is the smallest group which contains ##X##.

Lemma 4.2.: ##X \subseteq H \leq G \Longrightarrow F(X) \subseteq H##.
Lemma 4.3.: ##F(X) \leq G \,\wedge \,F(X)=gp(X)##.

Thank you. This helps.
A couple of points:

1) You say "Why should we restrict the statement? With respect of Lemma 4.3. it might be useful that it holds for any subgroup, G=HG=H included". My only answer is that find this material difficult and it helps if there are no
extraneous distractions. I even do things like use either all
⊇ or all
⊆.
Expressing everything in a parallel way makes it clearer to me.

2) I am not familiar with the term "
the words over
X".
I guess that means
{
x
ε1
1
...
x
εn
n
|
x
i
X,
ε
i
=±1,n
a positive integer
}?

Not sure what you mean by the subset signs.

If it is about meaning, then ##\subseteq## refers to sets, ##\leq## to subgroups, ##\trianglelefteq## to normal subgroups.

If you asked about my comment on Lemma 4.2. then I meant, that choosing subgroups ##H## instead of ##G## makes the statement easier for applications. If we have a situation (*) ##X \subseteq H \leq G## and want to apply it to ##H##, it is immediately clear. If Schaum had written it as ##X \subseteq G## and wanted to apply it for a situation (*), he would had been forced to complicatedly write: "Now we apply Lemma 4.2. with ##G=H##, but here at (*) we still have possibly ##H \subsetneq G## ..." which is a total mess. Generality in Lemma 4.2. as it stands is as good and easier to apply. And Lemmata are meant to apply on various situations.

Words in group theory are elements written as products and integer powers of certain group elements, called the alphabet. Here we have ##X=\{\,x_1,\ldots,x_n\,\}## as alphabet and words are any products or inverses of these.

## 1. What is Group Theory and why is it important?

Group Theory is a branch of mathematics that studies the properties of groups, which are sets of elements that follow specific rules of operation. It is important because it has applications in many areas of mathematics, physics, and chemistry, and helps us understand symmetry and patterns in nature.

## 2. What are some common Lemmas used in Group Theory?

Some common Lemmas used in Group Theory include the Lagrange's Theorem, which states that the order of a subgroup divides the order of the group, and the Cauchy's Theorem, which states that if a prime number divides the order of a group, then the group contains an element of that order.

## 3. How do I prove a Lemma in Group Theory?

To prove a Lemma in Group Theory, you need to show that the statement is true for all elements in the group. This can be done by using the axioms and properties of groups, such as closure, associativity, and identity element, and by using logical reasoning and mathematical induction.

## 4. What are some practical applications of Group Theory Lemmas?

Group Theory Lemmas have practical applications in various fields, such as cryptography, coding theory, and quantum mechanics. For example, the RSA algorithm, which is used for secure communication, is based on the properties of groups and uses the Lagrange's Theorem.

## 5. What is the best way to understand and remember Group Theory Lemmas?

The best way to understand and remember Group Theory Lemmas is to practice solving problems and proofs. It is also helpful to visualize the concepts and make connections with real-world applications. Additionally, discussing and teaching the Lemmas to others can also aid in understanding and retention.

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