Group Theory

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

• 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.

• \(o(g), \mathrm{ord}(g), o(G), |G|\)
• The order of an element \(g\in G\) is the smallest nonzero natural number \(n\) such that \[ g^n = e. \]
• The order of a group \(G\) is the number of elements within \(G\).
• For \(g\in G\), \(|g| \big| |G|\) by the Lagrange's theorem
• For a finite group \(G\) of order \(n\),
  • the number of elements in \(G\) with the order \(d\) elements in \(G\) is a multiple of, possibly zero times, \(\varphi(d)\),
  • where \(d\) is a divisor of \(n\) and \(\varphi\) is the Euler's totient function.

• 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.

• 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 \(G\).

• The order of the group \(G\) is equal to the sum of the size of its conjugacy classes: \[ |G| = |\mathrm{Z}(G)| + \sum_{\begin{smallmatrix}A \in G/\sim\\ A\not\subset \mathrm{Z}(G)\end{smallmatrix}}|G:\mathrm{C}_G(x(A))| \]
  • where \(x\) is a choice function that choose a single element of \(A\).
  • The equation is separating the trivial divisors and non-trivial divisors of \(|G|\):
    • \(|G:\mathrm{C}_G(x)|\) could be equalt to 1 if \(x\in \mathrm{Z}(G)\), and greater than 1 otherwise.
• Notice that \(|G:\mathrm{C}_G(x)| = |\mathrm{Cl}(x)|\) since each cosets of \(\mathrm{C}_G(x)\) corresponds to an element in the conjugacy class \(\mathrm{Cl}(x)\).
  • To wit, \(g\mathrm{C}_G(x)\) corresponds to \(gxg^{-1}\). It is well defined precisely because \(\mathrm{C}_G(x)\) is a centralizer.
  • In particular, the centralizer of \(x\), \(\mathrm{C}_G(x)\) corresponds to \(x\) itself.
• Also notice that each the elemnet in the center \(\mathrm{Z}(G)\) forms a conjugacy class by itself, since it is invariant under conjugation.
• The equation is counting the elements by conjugacy class, except for the center which is counted separately.
• It is also obtained by considering the group action of \(G\) on \(G\) by 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 like conjugation class within the power set.
• Analogous to the conjugacy class of elements:
  • \(|\mathrm{Cl}(S)| = |G:\mathrm{N}_G(S)|\)

• 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 \(\psi\colon G\to G^{\mathrm{op}}\).
  • \(\psi(g) = g^{-1}\) is one of them.

• Group action of a group \(G\) on a set \(S\) is a group homomorphism from \(G\) to some group of endomorphisms on \(S\).
• The group action is often curried, with the element of the group taken as the parameter.
  • \(\rho\colon G \to \mathrm{Aut}(X)\)
• Action on the other side can be constructed by composing with the inverse operation of the group.
  • \(\alpha'(x, g) = \alpha((g)^{-1}, x)\)
  • \[ \alpha'(x, gh) = \alpha(h^{-1}g^{-1}, x) = \alpha(h^{-1}, \alpha(g^{-1}, x)) = \alpha'(\alpha'(x,g), h) \]
• Action of a group on one side is the action of the opposite group on the other side.

• \[ |X/G| = \frac{1}{|G|}\sum_{g\in G}|X^g| \]
  • where \(G\) is a finite group acting on a set \(X\), and \(X^g\) is the left invarients of \( g \).
• For an infinite group \(G\), there is bijection:
  • \[ G\times X/G \longleftrightarrow \bigsqcup_{g\in G} X^g. \]
• The number of orbits is equal to the average number of points fixed by an element of \(G\).

• From the equivalent sum, \[ |\{(g,x)\in G\times X\mid g\cdot x = x\}| = \sum_{g\in G}|X^g| = \sum_{x\in X}|G_x| \]
• Notice that \[ \sum_{x\in X}|G_x| = \sum_{A\in X/G}|G| = |G|\cdot |X/G|. \]

• Necklace polynomial
• Circular permutation by bins
• Colorings of a Cube

• 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 \(g\), 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.
• The right conjugation of \(x\) by \(g\) is often denoted by \(x^g\).
• The inner automorphism group is denoted \(\mathrm{Inn}(G)\).
• Inner automorphism group \(\mathrm{Inn}(G)\) is a normal subgroup of the automorphism group \(\mathrm{Aut}(G)\).
  • Then quotient group is called outer automorphism group: \[ \mathrm{Out}(G) = \mathrm{Aut}(G)/\mathrm{Inn}(G) \]

• \(\mathrm{C}_G(S), \mathrm{N}_G(S) < G\)
  • By Lagrange's theorem, \(|\mathrm{C}_G(S)|, |\mathrm{N}_G(S)|\) divides \(|G|\).
• \(\mathrm{C}_G(S) \triangleleft \mathrm{N}_G(S)\)
• \(S\subseteq \mathrm{C}_G(S)\iff S\ \text{abelian}\)
• \(S\subseteq \mathrm{C}_G(\mathrm{C}_G(S))\)
  • If a subgroup \(H\) is self-bicommutant, \(H = \mathrm{C}_G(\mathrm{C}_G(H))\)
• \(H
• \(\mathrm{C_G(G)} = G \iff G\ \text{abelian}\)

• 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 \(H = G\))
  • \[ G/\mathrm{Z(G)} \cong \mathrm{Inn}(G). \]
  • From the homomorphism \(h\colon G\to \mathrm{Inn}(G)\colon g \mapsto (x\mapsto x^g)\), apply the first isomorphism theorem , with the center associated to the identity automorphism.

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.