http://bmw.byuimath.com/aa/ 2026-10-18T07:04:19+00:00 http://bmw.byuimath.com/aa/ http://bmw.byuimath.com/aa/lib/tpl/dokuwiki/images/favicon.ico text/html 2013-08-21T13:02:24+00:00 Anonymous (anonymous@undisclosed.example.com) http://bmw.byuimath.com/aa/doku.php?id=definition:abelian_group&rev=1377104544&do=diff Abelian Group Definition A group is if its binary operation is commutative; otherwise, it is . ---------- Remarks * None. ---------- $\LaTeX$ version %%%%% % DEPENDENCIES % RequiredMacros: \newcommand{\textdef}[1]{\textit{#1}} %%%%% \begin{definition} A group is \textdef{abelian} if its binary operation is commutative; otherwise, it is said to be \textdef{nonabelian}. \end{definition} text/html 2013-08-21T13:10:42+00:00 Anonymous (anonymous@undisclosed.example.com) http://bmw.byuimath.com/aa/doku.php?id=definition:alternating_group&rev=1377105042&do=diff Alternating Group Definition The $\textdef{alternating group}$ of degree $n$, or alternating group on $n$ letters, is the permutation group of even permutations on $n$ elements. We'll use the notation $A_n$ to denote the alternating group of degree $n$. ---------- Remarks * text/html 2013-08-22T16:28:37+00:00 Anonymous (anonymous@undisclosed.example.com) http://bmw.byuimath.com/aa/doku.php?id=definition:automorphism_group&rev=1377203317&do=diff Automorphism Group Definition Let be a set possibly equipped with additional structure, e.g. is the vertex set of a graph or is the underlying set of a group. The permutations of “preserving the additional structure” is called the of the structure, denoted text/html 2013-08-22T16:29:45+00:00 Anonymous (anonymous@undisclosed.example.com) http://bmw.byuimath.com/aa/doku.php?id=definition:automorphism_of_a_graph&rev=1377203385&do=diff Automorphism Of A Graph Definition An is a permutation of the set of vertices such that two vertices and form an edge if and only if and form an edge. The is the set of all automorphisms of the graph. If is a graph, its automorphism group is denoted text/html 2013-11-21T11:14:54+00:00 Anonymous (anonymous@undisclosed.example.com) http://bmw.byuimath.com/aa/doku.php?id=definition:composition_combination_of_permutations&rev=1385050494&do=diff Composition Combination Of Permutations Definition Let be a set, and be a set of permutations of . * If is a permutation of , we'll use exponential notation to express repeated composition of . This gives us and , etc. * We'll use negative exponents when we want to repeated apply an inverse, which gives us text/html 2013-09-19T15:07:51+00:00 Anonymous (anonymous@undisclosed.example.com) http://bmw.byuimath.com/aa/doku.php?id=definition:encryption_key&rev=1379617671&do=diff Encryption Key Definition An encryption key is an algorithm which specifies how a message is to be encoded. The encrypted message is often called ciphertext. The original message is called plain text. See Wikipedia for more information. ---------- Remarks text/html 2013-08-21T13:01:33+00:00 Anonymous (anonymous@undisclosed.example.com) http://bmw.byuimath.com/aa/doku.php?id=definition:group&rev=1377104493&do=diff Group Definition Let be a set, and let be a binary operation on . The structure is called a if the following hold. * For all one has . * There is a unique such that for all one has . * For all there is a unique such that . We usually simply write text/html 2013-08-21T10:01:29+00:00 Anonymous (anonymous@undisclosed.example.com) http://bmw.byuimath.com/aa/doku.php?id=definition:group_homomorphism&rev=1377093689&do=diff Group Homomorphism Definition Let $(G,\cdot)$ and $(H,\times)$ be groups. We say that the function $f:G\to H$ is a group $\textdef{homomorphism}$ if $f(a\cdot b)=f(a)\times f(b)$ for every $a,b\in G$. ---------- Remarks * I would like to add something along the lines of text/html 2013-08-21T10:00:19+00:00 Anonymous (anonymous@undisclosed.example.com) http://bmw.byuimath.com/aa/doku.php?id=definition:group_isomorphism&rev=1377093619&do=diff Group Isomorphism Definition Let be a homomorphism of groups. If is also a bijection, then we say that is a (group) . If there exists an isomorphism between and , then we say that and are , denoted . ---------- Remarks * Sometimes we may say that two groups are the same (or equal) when we mean that they are isomorphic. text/html 2013-08-21T10:03:32+00:00 Anonymous (anonymous@undisclosed.example.com) http://bmw.byuimath.com/aa/doku.php?id=definition:image_of_a_group_homomorphism&rev=1377093812&do=diff Image of a Group Homomorphism Definition Let be a group homomorphism. The of is the set of elements of that are mapped to by , namely ---------- Remarks * I would like to use Cryptography to help the students see that the image of a group homomorphism must always be a subgroup of text/html 2013-08-21T10:05:52+00:00 Anonymous (anonymous@undisclosed.example.com) http://bmw.byuimath.com/aa/doku.php?id=definition:kernel_of_a_group_homomorphism&rev=1377093952&do=diff Kernel of a Group Homomorphism Definition Let $f:G\to H$ be a group homomorphism. The $\textdef{kernel}$ of $f$ is the collection of elements of $G$ that map to the indentity $e_H$ element of $H$. In set notation, we write $$\ker f = \{ g\in G\mid f(g)=e_H\}.$$ ---------- text/html 2013-08-21T12:58:03+00:00 Anonymous (anonymous@undisclosed.example.com) http://bmw.byuimath.com/aa/doku.php?id=definition:order_for_groups&rev=1377104283&do=diff Order (for Groups) Definition Let be a group with identity , and let . * The of , denoted , is the cardinality of . * The of , denoted , is the smallest positive integer such that , if such an exists. If no such exists, we say has infinite order. text/html 2013-08-30T11:47:09+00:00 Anonymous (anonymous@undisclosed.example.com) http://bmw.byuimath.com/aa/doku.php?id=definition:permutation&rev=1377877629&do=diff Permutation of $X$ Definition Let be a set. A of is a bijection from to . We can think of a permutation of as a way of rearranging the elements in . The identity permutation is the permutation defined by . ---------- Remarks * This is used to define the text/html 2013-08-30T11:47:57+00:00 Anonymous (anonymous@undisclosed.example.com) http://bmw.byuimath.com/aa/doku.php?id=definition:permutation_group&rev=1377877677&do=diff Permutation Group Definition A $\textdef{permutation group}$ on $X$ is a set of permutations of $X$ that contains the identity permutation and is closed under function composition and taking inverses. ---------- Remarks * Alternately, a permutation group is a subgroup of $\sym{X}$, where $\sym{X}$ denotes the text/html 2013-09-27T16:38:58+00:00 Anonymous (anonymous@undisclosed.example.com) http://bmw.byuimath.com/aa/doku.php?id=definition:simple_shift_permutation_of_the_alphabet&rev=1380314338&do=diff Simple Shift Permutation Of The Alphabet Definition Let $M$ be a message, which we will think of as a sequence of letters from the standard 26 letter Roman alphabet. We can write $M=(m_1,m_2,m_3, \ldots, m_n)$ where $m_i\in\{a, b, c, \ldots, z\}$. For simplicity, our messages are not case-sensitive and all spaces and punctuation have been removed. To send the phrase text/html 2013-08-21T13:09:18+00:00 Anonymous (anonymous@undisclosed.example.com) http://bmw.byuimath.com/aa/doku.php?id=definition:subgroup&rev=1377104958&do=diff Subgroup Definition Let be a group, and let be a nonempty subset of . Then is called a of if the following hold: * for all one has , and * is a group. When is a subgroup of , we write . Any subgroup of that is not equal to itself is called a text/html 2013-08-14T10:08:58+00:00 Anonymous (anonymous@undisclosed.example.com) http://bmw.byuimath.com/aa/doku.php?id=definition:sylow_p-subgroup&rev=1376489338&do=diff Sylow $p$-subgroup Definition Let be a group and a prime. A $p$-subgroup is called a Sylow $p$-subgroup of if is not properly contained in any other $p$-subgroup of , i.e. if is a maximal -subgroup of . The collection of all Sylow -subgroups of is usually denoted . ---------- text/html 2013-08-22T16:47:31+00:00 Anonymous (anonymous@undisclosed.example.com) http://bmw.byuimath.com/aa/doku.php?id=definition:symmetric_group&rev=1377204451&do=diff Symmetric Group Definition Let be any set. The on , denoted , is the set of all permutations of . We denote by the symmetric group on . ---------- Remarks * None. ---------- $\LaTeX$ version %%%%% % DEPENDENCIES % RequiredMacros: \newcommand{\textdef}[1]{\textit{#1}} \DeclareMathOperator{\sym}{Sym} %%%%% \begin{definition} Let $X$ be any set. The \textdef{symmetric group} on $X$, denoted $\sym(X)$, is the set of all permutations of $X$. We denote by $S_n$ the symmetric group on … text/html 2013-08-15T12:42:15+00:00 Anonymous (anonymous@undisclosed.example.com) http://bmw.byuimath.com/aa/doku.php?id=definition:template&rev=1376584935&do=diff @!!PAGE@ Definition Type the definition using wiki syntax. ---------- Remarks * Put them in a bulleted list. ---------- $\LaTeX$ version %%%%%%%%%% % DEPENDENCIES % RequiredMacros: \DeclareMathOperator{\syl}{Syl} %%%%%%%%%% \begin{definition} Type the definition using LaTeX syntax. \end{definition} text/html 2013-11-25T13:35:40+00:00 Anonymous (anonymous@undisclosed.example.com) http://bmw.byuimath.com/aa/doku.php?id=definition:the_order_of_a_permutation&rev=1385404540&do=diff The Order Of A Permutation Definition The order of a permutation $\sigma$ is the smallest positive integer $n$ such that $\sigma^n$ equals the identity permutation. ---------- Remarks * ---------- $\LaTeX$ version %%%%% % DEPENDENCIES % RequiredMacros: %%%%% \begin{definition} The order of a permutation $\sigma$ is the smallest positive integer $n$ such that $\sigma^n$ equals the identity permutation. \end{definition}