More Groups

$D_n$ : Let $n \ge 3$, Dihedral group is a symmetrical regular n-gon denote by $D_n$

• The order of $D_n$ is $2n$
• Let $D_4$ be the group of all symmetries of the square with the operation being function composition. Denote $p_{n}$ by rotate n degree. Denote $r$ be reflection. Denote the square with side $a,b,c,d$. D means Dihedral

$Z_n$: the binary operations addition modulo $n$ on the set $\{0,1,\ldots,n-1\}$

• If $g$ is co-prime of $n$ for group $\Z_n$, then $\langle g \rangle = \Z_n$ is a cyclic group
• $\forall k, k|n \implies \langle n/k\rangle$ is unique subgroup of $\Z_n$ of order $k$
• Every infinite cyclic group is isomorphic to $\Z$
• Every finite cyclic group is isomorphic to some $\Z_n$

$U(n) = U_n$: the binary operations multiplication modulo $n$ on the set $\{0,1,\ldots,n-1\}$

$U_k(n)$: $\forall k|n, U_k(n) = \{x\in U(n) : x\mod k = 1\}$s

$S_n$: (Symmetric group) stand the group of injective functions from $\{1,\ldots,n\}$ to $\{1,\ldots,n\}$

• $S_3$ is the "same group" as $D_3$ by order
• $|S_n| = n!$
• even permutation in $S_n$ forms a subgroup of $S_n$
• $\sigma \in S_n, \sigma = (i_1j_1)\ldots(i_mj_m)$ then $sgn = 1$ when even, else odd
• sign function of multiplication $S_n$ is a homomorphism
• Any subgroup of $S_n$ acts on the $n-$element set $\{1,\ldots,n\}$

$A_n$: alternating group of degree $n$ or the group of even permutations of $n$ symbols

• $|A_n| = n!/2$

• $\forall n\le 3 \iff A_n$ is abelian

• commutator subgroup of $S_n$

• $GL_n(\R)$ or $GL(n, \R) = \{$all invertible real-valued $n\times n$ matrices$\}=\{$all invertible linear operators $\R^n\to \R^n$ matrices$\}$

• the determinant is a homomorphism

$SL_n(\R) = SL(n, \R) = \{A\in GL_n(\R) : det(A) = 1\}$

• $SL_n(\R)$ is the kernel of the determinant function map $GL_n(\R)\to \R ^{\times}$

$GL_n(\R) = GL(n, \R) = \{n\times n$ matrices of non-zero determinants with coefficients from the field $\R\}$