00.1.群の様々な定義

群の位数

$|G|$ を群の位数と呼ぶ.

群の元の位数

$a \in G$,
$\exists n$ such that $a^n = e$
then
$n$ を$a$の位数という.
$\nexists n$ such that $a^n = e$
then $a$の位数は $\infty$ とする.

群の生成元

$G$:Group, then
$\lt S \gt$ is defined as below
when
$S \subset H$, $H$ is a subgroup of $G$ $\Rightarrow \lt S \gt \subset H$
. すなわち
$\lt S \gt$は$S$を含む最小の部分群.
$\lt S \gt$ is called a minimal subgroup of $G$ including $S$.
別の定義1
$$ \lt S\gt = \bigcap_{\lambda \in \Lambda} G_\lambda \\ (G_\lambda)_{\lambda \in \Lambda} = \{G_\lambda|S \subset G_\lambda, G_\lambda:\text{subgroup of }G\} $$ 別の定義2
$G$ is a group.
$S = \{s_i \in G\} \subset G$.
$\lt S\gt$ is defined as
$\lt S\gt = \{s_0^{n_0}s_1^{n_1} \cdots s_i^{n_i} \}$.
when $\lt S\gt = G$, then "$S$ は $G$ を生成する"という.

同値関係・同値類

$S$:Set
同値関係"$\sim$"は次のように定義される.
for $\forall x, y \in S$
$x \sim y$ が成立するかしないかが一意に決まり
"$\sim$"が次の性質を持つ. for $\forall x, y, z \in S$
1. $x \sim x$
2. $x \sim y \Rightarrow y \sim x$
3. $x \sim y, y \sim z \Rightarrow x \sim z$
同値関係の要素を集めた$S$の部分集合で分けることができる.
これを同値類と呼び、同値類を集めた集合を $$S/\sim$$ と書く

剰余類(coset)

$G$:Group
$H$:Subgroup
$\sim$ is defined as
$a \sim b \Leftrightarrow a,b \in G, b^{-1}a \in H$
$\sim$ は同値関係
この同値関係によって得られる同値類全体を$H$による左剰余類(left coset)とよび
$G/H (= \{ aH | a \in G \} )$ と書く.

指数(index)

左剰余類$G/H$の数を$G$の$H$における指数(index)とよび
$|G/H| = (G:H)$ と書く

対称群・置換(Symmetric Group・Permutation)

$S_n$:n個の要素からn個の要素への全単射全体の集合.
$S_n = \{f|f:\{1,...,n\} \Rightarrow \{1,...,n\}, f:\text{全単射}\}$
$f, g \in S_n, x \in \{1,...,n\}$ then $f\cdot g (x) = g(f(x))$
$(S_n,\cdot)$ is a group.
$S_n$ is called a Symmetric group.
$f \in S_n$ is called a Permutation.

巡回置換・互換(Cycle・Transposition or 2-cycle)

$S_n$:n次対称群
$(i,j,k) \in S_n$
$(i,j,k)(i) \mapsto j$
$(i,j,k)(j) \mapsto k$
$(i,j,k)(k) \mapsto i$
$(i,j,k)(x) \mapsto x \quad (x \neq i,j,k)$
then
$(i,j,k)$ is called a Cycle.
Especially
$(i,j)$ is called a Transposition.

偶置換・奇置換(Permutation・Even Permutation・Add Permutation)

* 偶数の互換の積で表される置換を偶置換と呼ぶ.
* 奇数の互換の積で表される置換を奇置換と呼ぶ.

交代群(Alternating Group)

すべての偶置換を要素としてもつ集合は$S_n$の部分群になる.
$A_n = \{p|p \in S_n, p = c_1 c_2 \cdots c_{2i}, i \in \mathbb{N}, c_i:\text{互換}\}$

クラインの４群

$G:S_n$
$V = \{e, (1 \ 2)(3 \ 4), (1 \ 3)(2 \ 4), (1 \ 4)(2 \ 3) \}$
このとき$G$をクラインの４群と呼ぶ.

単純群

群$G$ の正規部分群が $\{e\}$ と $G$ のみの時、$G$ を単純群と呼ぶ.

巡回群

$g \in G$ and $\lt g \gt (=\lt \{g\} \gt) = G$
then
$G$ を "$g$ の巡回群"と呼ぶ.
また$g$ を "$G$ の生成元"と呼ぶ.

例

$\mathbb{Z} = \lt 1 \gt$
$n\mathbb{Z} = \lt n \gt$

正規部分群(Normal Subgroup)

$G$:Group
$N$:Subgroup of $G$
if
$\forall x \in G, xNx^{-1} \subset N$
then
$N$ is called 正規部分群(normal subgroup)
$x = y \Rightarrow yNy^{-1} \subset N$
$x = y^{-1} \Rightarrow y^{-1}N(y^{-1})^{-1} \subset N$
$\Leftrightarrow y^{-1}Ny \subset N$
$\Leftrightarrow N \subset yNy^{-1}$
$\therefore N = yNy^{-1}$

剰余群

$G$:Group
$N$:Normal Subgroup of $G$
then
$G/N$ is defined by below
$G/N = \{aN|a \in G\}$
$aN = a'N \Leftrightarrow a^{-1}a' \in N$
$aN \cdot bN = (a \cdot b)N$
(this definition is well defined.)
$\because aNbN = abNN = abN$

交換子(Commutator)

$G$:Group
$x,y \in G$
$[x,y]$ is defined by
$[x,y] = xyx^{-1}y^{-1}$
and called 交換子(Commutator)

交換子群

$G$:Groupに対して交換子全体で生成される群を交換子群と呼び
$D(G)$ と書く.
$D(G) = \lt \{[x,y]|x,y \in G\} \gt$

可解群

$G$:Group
if
$\exists n \in \mathbb{N}, D^n (G) = \{e\}$
then
G is called a solvable(可解群) group.
else
G is called a non-solvable(非可解群) group.

準同型写像

Supporse
$G,G'$:Group
$f:G \rightarrow G'$
$f:x \mapsto x'$
then
if $f(xy) = f(x)f(y)$ then
$f$ is called a homomorphism(準同型写像)
if $\forall x' \in G', \exists x, f(x) = x'$ then
$f$ is surjective(全射)
if $\forall x,y \in G, x \neq y \Rightarrow f(x) \neq f(y)$ then
$f$ is injective(単射)

直積

$G_1,\cdots,G_n$:Group
then
$G_1 \times \cdots \times G_n := \{(g_1,\cdots,g_n|g_1 \in G_1, \cdots, g_n \in G_n)\}$
$(g_1, \cdots, g_n)\dot (h_1,\cdots,h_n) := (g_1 h_1, \cdots, g_n h_n)$

射影(Projection)

$G_1,\cdots, G_n$:Group
$\text{Prj}_i:G_1 \times \cdots \times G_i \times \cdots \times G_n \rightarrow G_i$
$(x_1, \cdots, x_i, \cdots, x_n) \mapsto x_i$

入射(Injection)

$G_1,\cdots, G_n$:Group
$\text{Inj}_i:G_i \rightarrow G_1 \times \cdots \times G_i \times \cdots \times G_n$
$x_i \mapsto (e_1, \cdots, x_i, \cdots, e_n)$

群の作用

$G$:Group
$S$:Set
$x \in S$
$\cdot:G \times S \rightarrow S$
$(g,x) \mapsto g\cdot x \in S$
(1) $e \cdot x = x \quad$ ($e$ is identity of $G$)
(2) $g \cdot (h \cdot x) = (gh) \cdot x \quad (g,h \in G)$
then
$\cdot$ is defined as action of $G$ over $S$

軌道

$G$:Group
$S$:Set
$\cdot:G \times S \rightarrow S$:Group Action of $G$ over $x \in S$
$S$
then
$O_x = \{s \in S|\exists g \in G, \text{ s.t. } s = g \cdot x\}$
is called as an orbit of $x$

固定化群(or安定化群)(Stabilizer Group)

$G$:Group
$S$:Set
$\cdot:G \times S \rightarrow S$:Group Action of $G$ over $x \in S$
$G_x = \{g \in G| g \cdot x = x\}$
$G_x$ is called a Stabilizer Group.

中心化群(Centralizer)

$G$:Group
$S \subset G$
$Z(S) = \{g \in G| \forall s \in S, s g = g s\}$
then $Z(S)$ is called a Centralizer of $S$
$x \in G$
$Z(x) = \{g \in G|xg = gx\}$
then $Z(x)$ is called a Centralizer of $x$

正規化群(Normalizer)

$G$:Group
$S \subset G$
then $N(S) = \{g \in G | gSg^{-1} = S\}$
then $N(S)$ is called a Normalizer of $S$

共役類

$G$:Group
$\cdot:G \times G \rightarrow G$
$(g,x) \mapsto gxg^{-1}$
then
$C(x) = \{y \in G| y = gxg^{-1} \quad (g \in G)\}$