Question

Show that if \(S\) is a subgroup, then \(S^{2} \equiv S S=S\), and \(t S=S\) if and only if \(t \in S\). More generally, \(T S=S\) if and only if \(T \subset S\).

Short answer

After using the closure property of subgroups, and the mathematical properties of subsets, it was shown that if \(S\) is a subgroup, then \(S^{2} = S\), and \(t S=S\) if and only if \(t \in S\), and more generally \(T S=S\) if and only if \(T \subset S\).

Step-by-step solution

Identify subgroup S properties

Start by reminding some basic properties of a subgroup \(S\): all elements \(t\) of \(S\) satisfy the closure properties, meaning the product of any two elements in \(S\) belongs to \(S\), and each element \(t\) in \(S\) has an inverse in \(S\).

Prove \(S^2 = S\)

By closure, for any two elements \(t_1\) and \(t_2\) in \(S\), their product \(t_1 \cdot t_2\) is in \(S\), which shows \(S^2 \subset S\). Also, each element of \(S\) can be seen as the product of itself and the identity, which is in \(S\), so \(S \subset S^2\). Together, we have \(S^2 = S\).

Prove \(tS=S\) if and only if \(t \in S\)

Assume \(t \in S\). By using closure, \(t\cdot s\), for any \(s \in S\), is in \(S\), so \(tS \subset S\). The inverse of \(t\), say \(t^{'}\), is also in \(S\). Hence, \(t^{'} \cdot (tS) = S\), showing \(S \subset tS\). Therefore, we have \(tS=S\). The other way round, if \(tS = S\), by definition, \(t\) must be in \(S\), proving the claim.

Prove \(TS=S\) if and only if \(T \subset S\)

For any \(t \in T\) and \(s \in S\), since \(t \cdot s\) is in \(S\) by closure (since \(T \subset S\)), \(TS \subset S\). Entities \(s \in S\), could be considered as the product of the identity \(e \in T\) and \(s\), hence \(S \subset TS\). Thus \(TS = S\). Conversely, if \(TS = S\), for any \(t \in T\), the product of \(t\) and the identity in \(S\) is \(t\), therefore \(T \subset S\).

Key concepts

Subgroup Properties

A subgroup is a special subset of a group that carries with it a few important properties inherited from the group itself. To understand subgroups, think of them as a smaller group within a bigger group. The defining features of a subgroup are:

• It must contain the identity element of the group.
• It should be closed under the group operation. This means if you take any two elements from the subgroup and apply the group’s operation, you should end up with another element that’s also in the subgroup.
• Every element in the subgroup must have an inverse that’s also within the subgroup.

These properties ensure that subgroups maintain the 'group-like' structure, but on a smaller scale. They obey the same fundamental rules, which makes them incredibly useful for solving various mathematical problems.

Closure Property

The closure property is fundamental to the nature of both groups and subgroups. It states that for any two elements in a subgroup, their product—when operated upon by the group operation—must also be in the subgroup.

If you have a subgroup \( S \) and you pick any two elements \( a \) and \( b \) from \( S \), then \( a \, ext{and} \, b \) being in \( S \) ensures \( a \cdot b \) is in \( S \) too. This operation can be anything from addition to multiplication, depending on what kind of group you are dealing with.

Closure thus maintains that all repeated application of group operations within a subgroup does not take you "outside" of the subgroup. This is why, in the original equation showing \( S^2 = S \), closure helps ensure that the product of any elements in \( S \), as well as combinations of those products, remains within \( S \). This property is instrumental in subgroup proofs and verifications.

Inverse Elements

Inverse elements in group theory work just like the inverse operations you're used to: think of how addition has subtraction and multiplication has division as inverse operations. Each element of a subgroup must have an inverse element within the same subgroup.

If \( t \) is an element in subgroup \( S \), then there exists an element \( t' \) in \( S \) such that \( t \cdot t' = e \), where \( e \) is the identity element. This means if you combine any element with its inverse, you get back to the identity element, effectively "undoing" the operation.

For instance, in the equation \( tS = S \), the existence of an inverse ensures that elements combine in ways that allow the properties of subgroups to remain intact. If \( t \) didn't have an inverse in \( S \), the structure of the subgroup could break, as we wouldn't be able to "undo" group operations within the set.

Identity Element

The identity element is the neutral element in any group or subgroup that doesn't change any other element when used in the group operation. It is such a fundamental idea because it acts as the 'do nothing' element.

In additive groups, the identity element is 0, while in multiplicative groups, it's 1. For any element \( a \) in a subgroup, combining it with the identity, say \( e \), leads to \( a \cdot e = a \) or \( e \cdot a = a \). Essentially, the element 'remains itself'.

In the context of subgroups like \( S \), the identity element must be present. It's a core component in verifying other properties like closure and inverses. For example, in the equation \( TS = S \), ensuring that each subgroup contains the identity means that every element can act just as itself, maintaining consistency in subgroup operations.