lbwei space

Groups

#Abstract Algebra
2022/09/22

Table of Content

Structural Properties

Definition

A structural property of a binary structure is one that must be shared by any isomorphic structure.

e.g. (structural)

  • The set has \(4\) elements.
  • The operation is commutative.
  • The equation \(a ∗ x = b\) has a solution in \(S\) for all \(a, b \in S\).

isomorphism map 過去之後，仍然保留的性質。

e.g. (non-structural)

  1. The set \(S\) is a subset of \(\mathbb{C}\).（沒人說一定要在 \(\mathbb{C}\)）
  2. The number \(4\) is an element.

Problem

Show that the property that the binary operator is commutative is a structure property.

Solution

To prove the property, we have to show that if \((S, *)\) satisfies \(*\) is commutative, then for any \((S', *')\) isomorphic to \((S, *)\), it also satisfies that \(*'\) is commutative.

Assume \(*\) is commutative on \((S, *)\). Given \((S', *')\) isomorphic to \((S, *)\), there exists a isomorphism \(\phi: S \to S'\). Since \(\phi\) is onto, for any \(a', b' \in S'\), there exists \(a, b \in S\) such that \(\phi(a) = a', \phi(b) = b'\), respectively.

Then we have

\[a' *' b' = \phi(a) *' \phi(b) = \phi(a*b) = \phi(b*a) = \phi(b) *' \phi(a) = b' *' a'.\]

Hence \(*'\) is commutative on \((S', *')\), which implies the binary operator is commutative is a structure property. ◼

Identity Element

Definition

Let \((S, *)\) be a binary structure. An element \(e\) of \(S\) is an identity element for \(*\) if \(e * s = s * e = s\) for all \(s \in S\).

e.g.

  • \(0\) in \((Z, +)\)
  • \(\bar 1\) in \((\mathbb{Z}_n, ·)\)

Theorem (uniqueness)

A binary structure \((S, *)\) has at most one identity element. That is, if there is an identity element, then it is unique.

Proof（反證法，略）

Theorem (after mapped)

Suppose that \((S, *)\) has an identity element \(e\) for \(*\). If \(\phi: S \to S'\) is an isomorphism of \((S, *)\) with \((S', *')\), then \(\phi(e)\) is an identity element for \(*'\).

Proof

We need to show that

\[\phi(e) *' s' = s' *' \phi(e) = s'\]

for all \(s' \in S'.\)

用 isomporhism 的性質。

Groups

Definition (group)

A group \((G, *)\) is a set \(G\), with a binary operation \(*\), such that

  1. \(*\) is associative.
  2. There exists an identity element \(e \in G\) for \(*\)
  3. There is an inverse \(a'\) of \(a\) such that \(a' * a = a * a' = e\), for all \(a \in G\).

e.g. (a group)

  • \((\mathbb{Z}, +)\);
  • \((\mathbb{Z}_n, +)\);
  • \((M_{m\times n}(\mathbb{R}), +)\); …

第三例是 entry 為實數的 \(m \times n\) matrices。

e.g. (not a group)

  • \((\mathbb{Z}, \cdot)\), only \(a = \pm 1\) has an inverse;
  • \((\mathbb{Z}_n, \cdot)\), inverse of \(\bar 0\) doesn’t exist;
  • \((\mathbb{Z}_{>0}, +)\), no identity element; …

再舉一個大例子：

The set \(\text{GL}(n, \mathbb{R})\) of all invertible \(n \times n\) matrices under matrix multiplication is a group. (\(\text{GL}\) stands for general linear.)

要驗證 \(\text{GL}\) 是不是 group，除了定義的三個條件之外，還要確認 \(\text{GL}\) 是不是 closed under multiplication！（因為 \(\text{GL}\) 只是某 binary structure 的子集）

利用 \(\text{det}(AB) = \text{det}(A)\text{det}(B)\) 驗證。

Abelian Group

A group \((G, *)\) is abelian if its binary operation is commutative.

\(+\) 保留給 commutative binary operation，其他用 \(\cdot\) (multiplication)。(Notation)

Properties of Groups

Suppose \(G\) is a group.

Cancellation Law

\(ab = ac \implies b = c\), and \(ba = ca \implies b = c\), for all \(a, b, c \in G\).

用 \(a^{-1}\) 和 \(e\) 證明。

Unique Solution

\(ax = b\) and \(ya = b\) have unique solutions \(x\) and \(y\) in \(G\).

Proof \(\rm I\) (existence)

Let \(x = a^{-1}b\) …

Proof \(\rm II\) (uniqueness)

Suppose \(ax_1 = b = ax_2\) …

Unique identity element and inverse

\(a^{-1}\) and \(e\) is unique, for \(a \in G\).

基本上 uniqueness 都是用反證法。

Corollary

For all \(a, b \in G\), \((ab)^{-1} = b^{-1}a^{-1}\).