Abstract Algebra - Group

Operation

An operation \(*\) on \(A\) is a rule which assigns to each ordered pair \((a, b)\) of elements of \(A\) exactly one element \(a * b\) in \(A\).

Group

By a group we mean a set \(G\) with an operation \(*\) which satisfies the axioms:

1. \(*\) is associative.
2. There is an element \(e\) in \(G\) such that \(a * e = a\) and \(e * a = a\) for every element \(a\) in \(G\).
3. For every element \(a\) in \(G\), there is an element \(a^{-1}\) in \(G\) such that \(a * a^{-1} = e\) and \(a^{-1} * a = e\).

💡 Note that even though a group is not necessariy communicative, for axiom 2 and 3, identity and inverse of any element in \(G\) has to be communicative.

Different Groups

Abelian Group

If the communicative law holds in a group \(G\), then the group is called a communicative group, or an abelian group.

Semigroup

A set \(A\) with an associative operation. (There does not need to be an identity element, nor do elements necessarily have inverses.)

Symmetric Group

For any set \(A\), the group of all the permutations of \(A\) is called the symmetric group on \(A\), and it is represented by the symbol \(S_A\).

💡 For any positive integer \(n\), the symmetric group on the set \(\{ 1, 2, 3, ..., n\}\) is called the symmetric group on n elements, and is denoted by \(S_n\).

Basic Properties of Groups

Theorem 1

If \(G\) is a group and \(a, b, c\) are elements of \(G\), then

1. \(ab = ac\) implies \(b =c\)
2. \(ba = ca\) implies \(b = c\)

Left cancellation and right cancellation.

Theorem 2

If \(G\) is a group and \(a, b\) are elements of \(G\), then

\(ab = e\) implies \(a = b^{-1}\) and \(b = a^{-1}\)

Mutual inverse if the product equals identity.

Theorem 3

If \(G\) is a group and \(a, b\) are elements of \(G\), then

1. \((ab)^{-1} = b^{-1}a^{-1}\)
2. \((a^{-1})^{-1} = a\)

Subgroup

Let \(G\) be a group and \(S\) is a nonempty subset of \(G\). If

1. \(S\) is closed with respect to multiplication,
2. \(S\) is closed with respect to inverses,

then \(S\) is a subgroup of \(G\).

💡 \(S\) is called a subgroup because it is a group. The set containing only the identity, namely \(\{ e \}\), is the smallest subgroup of \(G\). \(G\) itself is the largest subgroup of itself.

Normal Subgroup

Let \(H\) be a subgroup of a group \(G\). \(H\) is called a normal subgroup (正规子群) of \(G\) if it is closed with respect to conjugates, that is, if

\[ \forall \ a \in H, x \in G, \ xax^{-1} \in H \]

💡 Note that according to this definition, a normal subgroup of \(G\) is any nonempty subset of \(G\) which is closed with respect to products, with repect to inverses, and with respect to conjugates.

Theorem 1

If \(H\) is a normal subgroup of \(G\), then \(aH = Ha\) for every \(a \in G\).

Theorem 2

Let \(H\) be a normal subgroup of \(G\). If \(Ha = Hc\) and \(Hb = Hd\), then \(H(ab) = H(cd)\).

Center of Group

The center of a group \(G\) is the normal subgroup \(C\) of \(G\) consisting of all those elements of \(G\) which commute with every element of \(G\).

Order of Group Elements

If there exists a nonzero integer \(m\) such that \(a^m = e\), then the order of the element \(a\) is defined to be the least positive integer \(n\) such that \(a^n = e\). If there does not exist any nonzeror integer \(m\) such that \(a^m = e\), we say that \(a\) has ordrer infinity.

Quotient Groups

Let \(G\) be a group and let \(H\) be a normal subgroup of \(G\). The set consists of all the cosets of \(H\) is denoted by the symbol \(G/H\).

\[ G/H = \{Hx : \forall \ x \in G \} \]

\(G/H\) with coset multiplication is a group. The group \(G/H\) is called the quotient group (or factor group) of \(G\) by \(H\).

💡 \(G/H\) is a homomorphic image of \(G\). The homomorphic function is \(f(x) = Hx\) and \(**f\) is called the natural homomorphism from \(G\) onto \(G/H\)**.

Cosets

Left and Right Coset

Let \(G\) be a group, and \(H\) a subgroup of \(G\). For any element \(a\) in \(G\), the symbol \(aH\) denotes the set of all products \(ah\), as \(a\) remains fixed and \(h\) ranges over \(H\). \(aH\) is called a left coset of \(H\) in \(G\). Similarly, \(Ha\) is called a right coset of \(H\) in \(G\).

Theorem 1

The family of all the cosets \(Ha\), as \(a\) ranges over \(G\), is a partition of \(G\).

Theorem 2

If \(Ha\) is any coset of \(H\), there is a one-to-one correspondence from \(H\) to \(Ha\).

Theorem 3 [Lagrange's Theorem]

Let \(G\) be a finite group, and \(H\) be any subgroup of \(G\). The order of \(G\) is a multiple of the order of \(H\).

Theorem 4

If \(G\) is a group with a prime number \(p\) of elements, then \(G\) is a cyclic group. Furthermore, any element \(a \neq e\) in \(G\) is a generator of \(G\).

💡 What is says is that there is (up to isomorphism) only one group of any given prime order \(p\). For example, the only group (up to isomorphism) of order \(7\) is \(\mathbb{Z}_7\), the only group of order \(11\) is \(\mathbb{Z}_{11}\).

Theorem 5

The order of any element of a finite group divides the order of the group.

\[ (G: H) = \frac{\text{order of } G}{\text{order of }H} \]

💡 If \(G\) is a group and \(H\) is a subgroup of \(G\), the index of \(H\) in \(G\) is the number of cosets of \(H\) in \(G\). We denote it by \((G: H)\).

Cauchy's Theorem

If \(G\) is a finite group, and \(p\) is a prime divisor of \(|G|\), then \(G\) has an element of order \(p\).