Group Theory

A group \( (G, \cdot) \) is a set \( G \) equipped with binary operation \( \cdot\colon G\times G \to G \), such that

• Associativity: \( \forall a,b,c \in G, (a\cdot b)\cdot c = a\cdot (b\cdot c) \)
• Identity Element: \( \exists e \in G, \forall g \in G, e\cdot g = g\cdot e = g \)
• Inverse Element: \( \forall g\in G, \exists g^{-1} \in G, g\cdot g^{-1} = g^{-1}\cdot g = e \)

They are the direct products of the cyclic groups of order \(p^n\)s.

• Coset of a subgroup \(H
• Two elements \(h_1,h_2\in G\) is in the same (left) coset of \(H\), if and only if \(h_1^{-1}h_2 \in H\).
  • Notice that the operation \(h_1^{-1}h_2\) cancels the \(g\) in \(gH\).

• \(|G:H|\), \([G:H]\), \((G:H)\)
• For a group \(G\) and its subgroup \(H\), the index \(|G : H|\) is the number of left cosets of \(H\) in \(G\), equivalently, the number of right cosets of \(H\) in \(G\).

• For a finite group \(G\) and its subgroup \(H\), \[ |G| = |G:H|\cdot|H|. \]
  • In particular, the order \(|g|\) of any element \(g\in G\) is a divisor of \(|G|\), since \(g\) generates a cyclic group of order \(|g|\).
• For infinite group, it can be taken as the cardinal numbers.

• The equivalence relation \[ h_1\sim h_2 \iff h_1^{-1}h_2 \in H \] forms a partition into cosets.
• Each cosets are of the same order by the bijection between them.
  • The property of a group: Every element has its inverse.
• The index \(|G:H|\) is defined to be the number of cosets.
• Therefore the statement is proven.

Group homomorphism is a map \( f\colon G \to H \) between two groups \( G, H \), that respect the group operations \( \cdot_G, \cdot_H \): \[ \forall g, h \in G, f(g\cdot_G h) = f(g) \cdot_H f(h). \]

Group homomorphism with inverse homomorphism: \[ \exists f^{-1}\colon H\to G, f^{-1}\circ f = 1_G \land f\circ f^{-1} = 1_H. \]

For a group \(G\), the function of the form \[ \varphi_g\colon G\to G \colon x\mapsto g^{-1}xg \] is called (right) conjugation by \(g\) (often denoted by \( x^g \)), which is an endomorphism of \(G\).

It is an automorphism since it has left and right inverse \(\varphi_{g^{-1}}\), therefore any automorphism that arises from conjugation is called an inner automorphism. They form a group called inner automorphism group \( \operatorname{Inn}(G) \). \(\mathrm{Inn}(G)\) is a normal subgroup of the automorphism group \(\mathrm{Aut}(G)\). Furthermore, Then quotient group is called outer automorphism group: \[ \mathrm{Out}(G) = \mathrm{Aut}(G)/\mathrm{Inn}(G). \]

• Left coset is equal to the right coset: \(gN = Ng\).
  • This can be recognized as the property of the identity element: \([g][e] = [e][g]\).

• Center \(\operatorname{Z}(G)\) of a group \(G\) is the set of elements that commute with every element of \(G\):
  • \[ \operatorname{Z}(G) := \{z\in G\mid \forall g\in G, zg = gz\}. \]
  • \(\mathrm{Z}\) form the German, Zentrum, meaning 'center'.

• Special Normal Subgroup

• Normal Interior
• Largest normal subgroup of a subgroup \(H\) of a group \(G\).
• The core of \(H\) with respect to a subset \(S\subseteq G\) is: \[ \mathrm{Core}_S(H) := \bigcap_{s\in S}s^{-1}Hs. \]

• For a prime \(p\), the \(p\)-core \(O_p(G)\) of a finite group \(G\) is the largest normal \(p\)-subgroup of \(G\).
• It is normal core of every Sylow \(p\)-subgroup \(\mathrm{Syl}_p(G)\) of the group \(G\).

Orbit of \( x \in X \) is given by: \[ G\cdot x := \{ g\cdot x \mid g\in G\}. \]

The quotient of the action is the set of all orbits of \( X \) under the action of \( G \), denoted by \( X / G \).

The stabilizer subgroup of \( G \) with respect to \( x \) is the set of all elements in \( G \) that fix \( x \): \[ G_x := \{ g\in G\mid g\cdot x = x\}. \]

Let \( g\in G \) be fixed. The left invariants by \( g \) is: \[ X^g := \{x\in X\mid g\cdot x = x\}. \]

• Burnside's Counting Theorem, Cauchy-Frobenius Lemma, Orbit-Counting Theorem.

It counts the number of orbits.

• A group is called \(k\)-transitive if the group action can map any sequence of length \(k\), into any sequence of length \(k\) respecting the order of the sequence.

• \( \mathrm{Cl}(g), [g] \)

The conjugacy class is the equivalence class under the equivalence relation of conjugation: \[ a\sim b \iff \exists g\in G, b = gag^{-1}. \]

Conjugation is an group action of \( G \) on \( G \) itself, in particular an inner automorphism. The centralizer of \( g\in G \) is the stabilizer and conjugacy classe is the orbit. By orbit-stabilizer theorem, \[ |G| = |\mathrm{C}_g||[g]|. \]

The order of the group \(G\) is equal to the sum of the size of its conjugacy classes: \[ |G| = |\mathrm{Z}(G)| + \sum_{[g], g \not\subset \mathrm{Z}(G)}|G:C_g|. \] Notice that each element in the center \(\mathrm{Z}(G)\) forms a conjugacy class by itself, since it is invariant under conjugation.

Two subsets \(S, T\subseteq G\) are conjugate to each other if there exists \(g\in G\) such that \(T = gSg^{-1}\).

The conjugacy class under this equivalence relation is denoted \(\mathrm{Cl}(S)\). It is the conjugacy class within the power set.

Analogous to the conjugacy class of elements: \[|\mathrm{Cl}(S)| = |G:\mathrm{N}_G(S)|.\]

• Commutant

The centralizer \(\mathrm{C}_G(S)\) (sometimes \(\mathrm{Z}_G(S)\), \( C_g \)) of a subset \(S\) in a group \(G\) is the set of elements of \(G\) that commute with every element of \(S\): \[ \mathrm{C}_G(S) := \{g\in G\mid \forall s\in S, gs = sg\}. \]

Equivalently, set of element \(g\) of \(G\) such that conjugation by \(g\) leaves each element of \(S\) fixed: \[ \mathrm{C}_G(S) := \{g\in G\mid \forall s\in S, gsg^{-1} = s\} \]

The normalizer \(\mathrm{N}_G(S)\) of a subset \(S\) in a group \(G\) is the set of elements of \(G\) that satisfy the weaker condition of leaving the set \(S\) fixed under conjugation: \[ \mathrm{N}_G(S) := \{g\in G\mid gS = Sg\} = \{g\in G \mid gSg^{-1} = S\}. \]

A subgroup \(H\) of a group \(G\) is called a self-normalizing subgroup of \(G\) if \(\mathrm{N}_G(H) = H\)

• C-Closed

A subgroup \(H\) of a group \(G\) is said to be self-bicommutant if \(H = \mathrm{C}_G(S)\) for some subset \(S\) of \(G\).

For a subgroup \(H\) of a gorup \(G\), \[ \mathrm{N}_G(H)/\mathrm{C}_G(H) \cong F < \operatorname{Aut}(H). \]

Corollary (For \(H = G\)) \[ G/\mathrm{Z(G)} \cong \mathrm{Inn}(G). \]

Consider the homomorphism: \(h\colon G\to \mathrm{Inn}(G)\colon g \mapsto (x\mapsto x^g)\), and apply the first isomorphism theorem, with the center associated to the identity automorphism.

For a group \((G, * )\), the opposite group is \(G^{\mathrm{op}} = (G, * ')\) with \(g_1*'g_2 = g_2*g_1\).

Any antiautomorphism: \(\psi(g*h) = \psi(h)*\psi(g)\) is an isomorphism \(\psi\colon G\to G^{\mathrm{op}}\). In particular, \(\psi(g) = g^{-1}\).

Given a simpler subgroup and a quotient group, in how many ways can we combine these to form other, potentially more complex, groups.

Every group is isomorphic to a subgroup of a symmetric group.