Question

(a) Show that the only element \(a\) in a group with the property \(a^{2}=a\) is the identity. (b) Now use \(e_{G} \star e_{G}=e_{G}\) to show that any homomorphism maps identity to identity. (c) Show that if \(f: G \rightarrow H\) is a homomorphism, then \(f\left(g^{-1}\right)=[f(g)]^{-1}\).

Short answer

The proofs show that: (1) the only element in a group that squares to itself is the identity element, (2) any homomorphism maps identity to identity, and (3) if \(f: G \rightarrow H\) is a homomorphism, then \(f(g^{-1})=[f(g)]^{-1}\).

Step-by-step solution

Proof: The only element a in a group G with the property \(a^{2}=a\) is the identity

Let the element \(a\) be in group G. Given that \(a^{2}=a\), post-multiplying each side by \(a^{-1}\) (the inverse of \(a\) in group G), we get \(a \cdot a \cdot a^{-1} = a \cdot a^{-1}\). Then, by associativity, \((a \cdot a) \cdot a^{-1} = a \cdot e_G\), where \(e_G\) is the identity element in G. This simplifies to \(a \cdot a^{-1} = e_G\), therefore, we have that \(a = e_G\).

Proof: Any homomorphism maps identity to identity

Assume we have a group G and H and \(f: G \rightarrow H\) is a homomorphism. Let \(e_G\) and \(e_H\) be the identity elements in group G and H respectively. Then, \(f(e_G \star e_G) = f(e_G)\). By the property of homomorphism, this can be equivalently written as \(f(e_G) \star f(e_G) = f(e_G)\), which by Step 1, shows that \(f(e_G) = e_H\). Therefore, any homomorphism maps identity to identity.

Proof: If \(f: G \rightarrow H\) is a homomorphism, then \(f\left(g^{-1}\right) = [f(g)]^{-1}\

Assume we have a group G and H and \(f: G \rightarrow H\) is a homomorphism. Let g be an element in group G. We know that \(f(g \star g^{-1}) = f(e_G)\). This can be written as \(f(g) \star f(g^{-1}) = e_H\). It follows that \(f(g^{-1}) = [f(g)]^{-1}\), the inverse in H of \(f(g)\). This shows that if \(f: G \rightarrow H\) is a homomorphism, then \(f(g^{-1})=[f(g)]^{-1}\).

Key concepts

Group Theory

Understanding group theory is essential when studying algebraic structures. A group is a set of elements combined with an operation that satisfies four key properties: closure, associativity, the existence of an identity element, and the existence of inverse elements for each element in the group. Closure means that if you perform the group operation on any two elements in the group, you'll still end up with an element from the group. Associativity ensures that when performing the operation on three elements, the order in which pairs of operations are performed does not matter. Group theory not only pertains to abstract algebra but also has practical applications in fields like physics and chemistry, where symmetry plays a crucial role.

Using group theory's fundamental definitions and properties, mathematicians can explore the behavior of complex systems through a simpler set of rules. This is why proving specific characteristics of group elements and functions between groups—like homomorphisms—can lead to broader applications across various mathematical disciplines.

Identity Element

An identity element in group theory is a central concept where each group possesses a unique element that, when combined with any element of the group, yields that same element. This property must hold for every element within the group. In mathematical terms, if we denote the identity element as e, and an arbitrary element as a, then a * e = a and e * a = a, for all a in the group. The identity element can be thought of as a neutral or do-nothing element since its presence in an operation doesn't change the other element.

From the given exercise, we use the identity element to prove an important property about homomorphisms: they always map the identity element of one group to the identity element of another. It's crucial to pinpoint the identity element in these scenarios because it helps retain the structure of the group under the mapping by a homomorphism.

Inverse Elements

Within group theory, the idea of inverse elements is one that ensures the group's ability to 'undo' any action. For every element a in a group, there exists an inverse element a^-1 such that a * a^-1 = e, where e is the identity element. This property must be true for each element within the group. The inverse can be thought of as an opposite action; if you combine an element with its inverse, you end up back at the identity element.

The concept of inverse elements becomes practical in our exercise when proving that a group homomorphism correctly translates inverses from one group to another. When a homomorphism behaves this way, it preserves an essential aspect of the group's structure, which is how elements interact with their inverses, thus maintaining the underlying algebraic relationships.

Mathematical Proofs

Mathematical proofs are vital to understanding and confirming various properties within group theory. By constructing a logical argument that starts from known truths and reasoning to a conclusion, one can show that a statement is true beyond any doubt. In group theory, proofs are used to demonstrate the properties of elements and operations, such as showing that a function is a homomorphism or that an element is the identity.

For example, the exercise's Step 1 provides a proof showing that any element that squares to itself must be the identity. In Steps 2 and 3, proofs are constructed to show that homomorphisms preserve identities and inverses, which are essential aspects of group structure. Mathematical proofs require a clear understanding of definitions and theorems. They form the backbone of mathematical rigor and help to establish the truths that mathematicians, scientists, and engineers rely on.