Question

Let \(X\) be a subset of a group \(G\). A word on \(X\) is an element \(w\) of \(G\) of the form $$ w=x_{1}^{e_{1}} x_{2}^{e_{2}} \cdots x_{n}^{e_{n}} $$ where \(x_{i} \in X\) and \(e_{i}=\pm 1\). Show that the set of all words on \(X\) is a subgroup of \(G\).

Short answer

In conclusion, the set of all words on \(X\) satisfies all three criteria for being a subgroup of \(G\), i.e., it is non-empty, it is closed under the group operation, and each element's inverse is in the set. Therefore, the set of all words on \(X\) is indeed a subgroup of \(G\).

Step-by-step solution

Non-Empty Subset

Let's first ensure that our set is not empty. Given that for every element \(x \in X\), its inverse \(x^{-1}\) must also be in \(X\). So, the identity element of the group \(G\), denoted by \(e_G\), can be expressed as \(x x^{-1}\), and therefore, is a word on \(X\). This ensures our set is non-empty.

Closure Property

Now, let's show that our set is closed under the group operation. Let \(w_1\) and \(w_2\) be two arbitrary words on \(X\). Since group \(G\) is closed under the operation, \(w_1 \cdot w_2\) is an element of \(G\) and it can be seen as a new word on \(X\). Thus, the set of all words on \(X\) satisfies the closure property.

Inverse element

Finally, we need to show that for each word in \(X\), its inverse is also a word in \(X\). For any word \(w\) on \(X\), \(w = x_{1}^{e_{1}} x_{2}^{e_{2}} \cdots x_{n}^{e_{n}}\), its inverse \(w^{-1}\) within \(G\) can be expressed as \(x_{n}^{-e_{n}} x_{n-1}^{-e_{n-1}} \cdots x_{1}^{-e_{1}}\). As \(x_{i}^{-1} \in X\) for \(x_{i} \in X\), \(w^{-1}\) is also a word on \(X\). Therefore, the set of all words on \(X\) contain their own inverses.

Key concepts

Group Theory

Group theory is a cornerstone of abstract algebra that studies the algebraic structures known as groups. A group is defined as a set equipped with a single binary operation that combines any two elements to form a third element while adhering to four fundamental rules: closure, associativity, the presence of an identity element, and the presence of inverse elements for all elements within the group. In our exercise, the group in question is denoted by 'G', and we examine a subset of this group to determine if it also forms a group, also known as a subgroup.

In practice, recognizing subgroups is essential not just in algebra, but also across mathematics and science since they reveal underlying symmetries and structure. For example, understanding the subgroup structure of a chemical group can provide insights into the possible reactions and molecular transformations. In physics, subgroups can highlight symmetries that underpin fundamental forces and particles.

Closure Property

The closure property is an indispensable aspect of group theory, playing a crucial role in defining a group. It is essentially the requirement that performing the group operation on any two elements of the set results in an element that is also within the set. This property ensures that the group is 'closed' under the operation; there are no surprises or undefined behavior when we combine elements.

In the textbook solution, we examined this property by considering any two words formed from a certain subset 'X' of the group 'G'. By demonstrating that the group operation, when applied to any two such words, results in another word that also belongs to the set, we affirm that the set satisfies the closure property.

• For example, if we take two elements, say 'a' and 'b', from a set that possess the closure property, we are guaranteed that 'a*b' (where '*' denotes the group operation) also belongs to the set.

Inverse Elements

The concept of inverse elements is integral to the structure of a group in group theory. An inverse element is one that, when combined with the original element under the group operation, yields the identity element of the group. In simpler terms, for any element 'a' in a group, there exists an inverse denoted as 'a^{-1}', such that the operation of 'a' with 'a^{-1}' brings us back to the basic identity element of the group.

This property forms one of the subgroup criteria in our exercise. For a set to be a subgroup, every element must have an inverse within the set. The exercise shows that the inverse of a word constructed from subset 'X' is also a word on 'X', which is in line with the inverse property of subgroups. The inverses ensure a kind of balance within the group, meaning that for every operation, there is a way to undo or reverse the effect and return to the starting, or identity, element.

• For instance, in the context of rotational symmetry, the rotation of an object clockwise by a certain angle has an inverse operation: rotating it counterclockwise by the same angle.