#LyX 1.6.2 created this file. For more info see http://www.lyx.org/
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\author "" 
\author "" 
\end_header

\begin_body

\begin_layout Description
Group Set 
\begin_inset Formula $G$
\end_inset

 and binary operation 
\begin_inset Formula $\cdot$
\end_inset

.
\end_layout

\begin_deeper
\begin_layout Itemize
Closure 
\begin_inset Formula $a\cdot b\in G$
\end_inset


\end_layout

\begin_layout Itemize
Associativity 
\begin_inset Formula $(a\cdot b)\cdot c=a\cdot(b\cdot c)$
\end_inset


\end_layout

\begin_layout Itemize
Identity 
\begin_inset Formula $e\cdot a=a\cdot e=a$
\end_inset


\end_layout

\begin_layout Itemize
Inverse 
\begin_inset Formula $a\cdot a^{-1}=a^{-1}\cdot a=e$
\end_inset


\end_layout

\end_deeper
\begin_layout Description
Identity is unique
\end_layout

\begin_layout Description
Order 
\begin_inset Formula $|x|=\min k\textrm{ s.t. }\left(x^{k}=e\right)$
\end_inset


\end_layout

\begin_layout Description
Direct
\begin_inset space ~
\end_inset

product Set:
\begin_inset Formula $A\times B$
\end_inset

 Operation:
\begin_inset Formula $\left(a_{1},b_{1}\right)\left(a_{2},b_{2}\right)=\left(a_{1}a_{2},b_{1}b_{2}\right)$
\end_inset


\end_layout

\begin_layout Description
Subgroup 
\begin_inset Formula $H\leq G$
\end_inset

, 
\begin_inset Formula $hk\in H$
\end_inset

 and 
\begin_inset Formula $h^{-1}\in H$
\end_inset


\end_layout

\begin_deeper
\begin_layout Description
Normal 
\begin_inset Formula $H\trianglelefteq G$
\end_inset

 , if 
\begin_inset Formula $gHg^{-1}=H$
\end_inset

 for all 
\begin_inset Formula $g\in G$
\end_inset


\end_layout

\begin_layout Description
Coset left: 
\begin_inset Formula $gH$
\end_inset

 right: 
\begin_inset Formula $Hg$
\end_inset

 with representative 
\begin_inset Formula $g$
\end_inset


\end_layout

\end_deeper
\begin_layout Description
Conguate of 
\begin_inset Formula $n\in N$
\end_inset

 by 
\begin_inset Formula $g$
\end_inset

.
 
\begin_inset Formula $gng^{-1}$
\end_inset


\end_layout

\begin_layout Description
Abelian
\begin_inset space ~
\end_inset

Group 
\begin_inset Formula $a\cdot b=b\cdot a$
\end_inset

 
\end_layout

\begin_layout Description
Homomorphism
\begin_inset Formula $\varphi:G\rightarrow H$
\end_inset

 such that 
\begin_inset Formula $\varphi(xy)=\varphi(x)\varphi(y)$
\end_inset


\end_layout

\begin_deeper
\begin_layout Itemize
Kernel: 
\begin_inset Formula $\varphi^{-1}\left(e_{H}\right)$
\end_inset


\end_layout

\begin_layout Itemize
Isomorphism 
\begin_inset Formula $G\cong H$
\end_inset

, a bijective homomorpism
\end_layout

\begin_layout Itemize
Automorphism: an isomorphism from 
\begin_inset Formula $G\rightarrow G$
\end_inset


\end_layout

\begin_layout Description
Mod 
\begin_inset Formula $G_{/H}:=\{\sigma H\mid\sigma\in G\}$
\end_inset

, set of equivalence classes of 
\begin_inset Formula $G$
\end_inset

 under the equivalence relation defined by 
\begin_inset Formula $\varphi$
\end_inset

.
 Is a group iff 
\begin_inset Formula $H$
\end_inset

 is normal
\end_layout

\end_deeper
\begin_layout Description
Field 
\begin_inset Formula $\left(F,+\right)$
\end_inset

 and 
\begin_inset Formula $\left(F-\{0\},\cdot\right)$
\end_inset

 are abelian groups and
\end_layout

\begin_deeper
\begin_layout Itemize
\begin_inset Formula $a\cdot(b+c)=(a\cdot b)+(a\cdot c)$
\end_inset


\end_layout

\end_deeper
\begin_layout Description
Group
\begin_inset space ~
\end_inset

Action on set 
\begin_inset Formula $A$
\end_inset

 is map 
\begin_inset Formula $G\times A\rightarrow A$
\end_inset


\end_layout

\begin_deeper
\begin_layout Itemize
\begin_inset Formula $g_{1}\cdot\left(g_{2}\cdot a\right)=\left(g_{1}g_{2}\right)\cdot a$
\end_inset


\end_layout

\begin_layout Itemize
\begin_inset Formula $1\cdot a=a$
\end_inset


\end_layout

\end_deeper
\begin_layout Description
Stabilizer 
\begin_inset Formula $G$
\end_inset

 acts on 
\begin_inset Formula $A$
\end_inset

,
\begin_inset Formula $\{g\in G\mid ga=a\}$
\end_inset


\end_layout

\begin_layout Paragraph
Examples of Groups:
\end_layout

\begin_layout Itemize
Symmetric 
\begin_inset Formula $S_{n}=\zeta_{n}$
\end_inset

 bijections of 
\begin_inset Formula $\{1,2,...,n\}$
\end_inset


\end_layout

\begin_layout Itemize
Dihedral 
\begin_inset Formula $D_{2n}$
\end_inset


\end_layout

\begin_layout Itemize
Quaterion 
\begin_inset Formula $Q_{8}$
\end_inset


\end_layout

\begin_layout Itemize
General Linear Group of degree n 
\begin_inset Formula $GL_{n}(F)$
\end_inset

, 
\begin_inset Formula $\left\{ A\mid A\textrm{ is an }n\times n\textrm{ matrix whse entries come from }F\textrm{ and }\det(A)\neq0\right\} $
\end_inset


\end_layout

\begin_layout Itemize
Integers Modulo n 
\begin_inset Formula $\mathbb{Z}/n\mathbb{Z}$
\end_inset


\end_layout

\begin_layout Itemize
Special Unitary Group 
\begin_inset Formula $SU(1,1)=\left\{ \left(\begin{array}{cc}
\alpha & \beta\\
\overline{\beta} & \overline{\alpha}\end{array}\right)\mid\alpha,\beta\in\mathbb{C},|\alpha|^{2}-|\beta|^{2}=1\right\} $
\end_inset


\end_layout

\begin_layout Itemize
Projective Special Unitary Group 
\begin_inset Formula $PSU(1,1)=SU(1,1)/\left\{ \pm\left(\begin{array}{cc}
1 & 0\\
0 & 1\end{array}\right)\right\} $
\end_inset


\end_layout

\begin_layout Description
Ring a set with binary operations 
\begin_inset Formula $+$
\end_inset

, 
\begin_inset Formula $\times$
\end_inset


\end_layout

\begin_layout Itemize
\begin_inset Formula $(R,+)$
\end_inset

 is an abelian group
\end_layout

\begin_layout Itemize
\begin_inset Formula $\times$
\end_inset

 is associative: 
\begin_inset Formula $(a\times b)\times c=a\times(b\times c)$
\end_inset


\end_layout

\begin_layout Itemize
distribution of multiplication over addition:
\end_layout

\begin_deeper
\begin_layout Itemize
\begin_inset Formula $(a+b)\times c=(a\times c)+(b\times c)$
\end_inset


\end_layout

\begin_layout Itemize
\begin_inset Formula $a\times(b+c)=(a\times b)+(a\times c)$
\end_inset


\end_layout

\end_deeper
\begin_layout Itemize
usually, contains identity: 
\begin_inset Formula $1\times a=a\times1=a$
\end_inset

 for all 
\begin_inset Formula $a\in R$
\end_inset


\end_layout

\begin_layout Itemize
called commutative if multipcation commutes
\end_layout

\begin_layout Itemize
If 
\begin_inset Formula $R/\{0\}$
\end_inset

 is a group under 
\begin_inset Formula $\times$
\end_inset

, R is a field
\end_layout

\begin_layout Paragraph
Ideal
\end_layout

\begin_layout Itemize
\begin_inset Formula $I$
\end_inset

 is an abelian subgroup of 
\begin_inset Formula $R$
\end_inset


\end_layout

\begin_layout Itemize
\begin_inset Formula $r\times a\in I$
\end_inset

 for all 
\begin_inset Formula $r\in R$
\end_inset

 and 
\begin_inset Formula $a\in I$
\end_inset


\end_layout

\begin_layout Paragraph
Quotient Ring
\begin_inset Formula $R/I=\{r+I|r\in R\}$
\end_inset


\end_layout

\begin_layout Description
Prime
\begin_inset space ~
\end_inset

Ideal 
\begin_inset Formula $I$
\end_inset

 is prime iff 
\begin_inset Formula $R/I$
\end_inset

 is an integral domain
\end_layout

\begin_layout Description
Zero
\begin_inset space ~
\end_inset

disizor 
\begin_inset Formula $a$
\end_inset

 is a zero divisor if there exists a nonzero
\bar under
 
\begin_inset Formula $b$
\end_inset

 
\bar default
such that 
\begin_inset Formula $ab=0$
\end_inset


\end_layout

\begin_layout Itemize
a ring with no zero divisors is called an 
\series bold
integral domain
\end_layout

\begin_layout Description
Maximal
\begin_inset space ~
\end_inset

Ideal 
\begin_inset Formula $M$
\end_inset

 is maximal iff 
\begin_inset Formula $A/M$
\end_inset

 is a field
\end_layout

\end_body
\end_document