Group Theory Note

#grouptheory #math #note #lecture

1.1 Symmetry

Naive concept about group: set of element with symmetrical operation.

1.2 Definition of Group

A group is a set with operation:

• $G\times G \to G$ i.e. $g_1 \cdot g_2 \to g_1g_2$

Axioms:

• $\forall g_1, g_2, g_3 \in G$, $(g_1 g_2) g_3 = g_1(g_2 g_3)$ (associativity)
• $\exists e \in G$, $g \cdot e = e \cdot g = g$ (identity)
• $\forall g \in G$, $\exists g’ \in G$, where $g’g = e$, denote $g’$ as $g^{-1}$ (inverse)

About triangle, we can define our group as: G: {e, R, R^2, S_A, S_B, S_C}

take it as example, we can see the following:

• $S_A^{-1} = S_A$
• $S_B^{-1} = S_B$
• $S_C^{-1} = S_C$
• $R^{-1} = R^2$
• $(R^2)^{-1} = R2$

1.3 Corollaries of the axioms

• $e_1 = e_1e_2 = e_2$ (unique unit)
• $g’g = e$, $g’‘g = e$, $g’‘g’g = g’’e = g’’$, $g’g’‘g = g’e = g’$, $g’ = g’’$(unique inverse)
• $\forall g \in G$, $(g^{-1})^{-1} = g$
• $\forall g_1, g_2 \in G$, $g_1h = g_2h \implies g_1 = g_2$
• $xg_1 = g_2$ and $g_1 x = g_2$ have unique solution, $g_2g_1^{-1}$

Abellain

Abellian: A group is called “Abellian”, if $\forall g_1, g_2 \in G$, we have $g_1g_2 = g_2g_1$ Example:

• (Q, +), (R, +)
• (C, +), (F, +), (V, +) (C, F, V stand for complex number, field, vector space respectively)
• the multiplication on Q, C, R, F without 0.

They are abellian(g1g2 = g2g1)

The cyclic Group, $\mathbb{Z}/n\mathbb{Z}$ is abellian, too.

1.5 Order, Subgroup, Isomorphism

Order of Group

We denote the order of G, as |G|, which is the number of operation in G. $|\mathbb{S}_n| = n!$, $|\mathbb{G}| \in \mathbb{N} \cup \inf$
we say the order of G can be identified as either finite or infinite.

Order of Element: An order ‘k’ of element is the smallest $k \in \mathbb{N}$, s.t. $g^k = e$, e.g. $e, g, g^2, g^3,g^4,\cdots,e$ $\text{Ord}{g} \in \mathbb{N} \cup {\inf}$

In Z, ord(0) = 1, Ord(k) = inf, …

Subgroup

Def: G is a group, H is subset of G, H is a subgroup, if

• Closed, $\forall h_1, h_2 \in H$, $h_1 h_2 \in H$
• $e \in H$
• $\forall h \in H$, $h^{-1} \in H$. Operation of H is the same as G.

Some corollaries:

• $H_1, H_2$ is subgroup of G, implies $H_1 \cap H_2$ is subgroup, too.
• $\forall G$, G and {e} are subgroup of G.

Example of Subgroups:

• G = Z, H is even Number.
• G = Z/10 Z, H = {0, 5}
• S, H = {e, (1 2 3, 2 3 1), (1 2 3, 3 1 2)}

Isomorphism

f: G_1 -> g_2 f is isomorphism, if $f(g_1g_2) = f(g_1) f(g_2)$, here $g_1 g_2 \in G_1$, $g_1, g_2 \in G_2$. And f is bijection.

• Composition of isomorphism is an isomorphism.
• Inverse of an isomorphism is also an isomorphism.

Lagrange Theorem

Theorem: Let G be a finite group, for $a \in G$, we have $a^{|G|} = e$.

|G| is the order of G, which stands for the number of elements in G. Moreover, |G| can be divided by ord(a).