Abstract Algebra IBL Wiki problem

problem:a_b_sqrt_2

Anonymous (anonymous@undisclosed.example.com) — 2013-12-24T22:59:17+00:00

A B Sqrt 2 Problem Let be the set . Please show * that is a group under addition. We will denote this group . * and that is abelian * that is a group under multiplication. We will denote this group as . * ---------- Remarks * There are more properties, I will add them later. I'm going to add all the properties of a field.

problem:aut-square

Anonymous (anonymous@undisclosed.example.com) — 2013-08-22T16:30:40+00:00

Automorphism group of a square Problem Consider the graph drawn below. The vertex set is and the (symmetric) relation giving adjacency is . Specifically, and Write down all elements of ; view these as the elements of that preserve . Also, determine if the following statements are true or false; explain your answers. For

problem:automorphisms_of_a_directed_square

Anonymous (anonymous@undisclosed.example.com) — 2013-09-27T17:20:39+00:00

Automorphisms Of A Directed Square Problem Consider the directed graph shown below with vertex set and arrows (directed edges) %center% * How would you define an automorphism of a directed graph? * List all the automorphisms of this directed graph. You should have 4.

problem:automorphisms_of_a_square

Anonymous (anonymous@undisclosed.example.com) — 2013-09-27T17:11:45+00:00

Automorphisms Of A Square Problem Consider the graph drawn below. The vertex set is and the set of edges is %center% Write down all the automorphisms of (there are more than 4, but less than 10). Explain how you know you have listed every automorhpism of

problem:automorphismsofasquare2

Anonymous (anonymous@undisclosed.example.com) — 2013-09-27T17:17:51+00:00

Automorphismsofasquare2 Problem Again consider the graph $\mathcal{G} = (V,E)$ shown below with vertex set $V = \{1,2,3,4\}$ and edges $$E = \{\{1,2\},\{2,3\},\{3,4\},\{1,4\}\}.$$ %center% Recall that set of all automorphisms of $\mathcal{G}$ is written $\aut(\mathcal{G})$. We listed all the elements in this set in problem

problem:cryptography_reading_assignment

Anonymous (anonymous@undisclosed.example.com) — 2013-09-27T18:45:51+00:00

Cryptography Reading Assignment Problem Download the free open source book Abstract Algebra: Theory and Applications by Thomas W. Judson. See . - Read pages 103-110. This is chapter 7, Introduction to Cryptography. It's OK if you don't understand all the notation. - In your preparation log, ask at least three questions about parts of the reading you would like to understand more about. It's OK if you don't understand all of the reading. He uses some theorems and …

problem:diagramatic_representation_of_sn

Anonymous (anonymous@undisclosed.example.com) — 2013-08-22T16:35:13+00:00

Diagramatic Representation Of $S_n$ Problem The goal of this problem is to find a way to represent the symmetric group with diagrams. We will focus on . * Which element of does the following diagram seem to represent? * What is the diagram for the inverse of the previous element.

problem:do_we_need_the_associative_law

Anonymous (anonymous@undisclosed.example.com) — 2013-11-21T10:32:55+00:00

Do We Need The Associative Law Problem In this problem, your job is to explain each step in the process of solving for . Try to break each part of your computation down into a the most basic process. As always, write your solution using complete sentences.

problem:examples_of_abstract_groups

Anonymous (anonymous@undisclosed.example.com) — 2013-08-21T09:50:20+00:00

Examples Of Abstract Groups Problem Give examples of groups with the following properties. Feel free to make them as simple as possible. * A finite abelian group * A finite nonabelian group * A finite group with exactly one proper nontrivial subgroup * An infinite abelian group * An infinite nonabelian group *

problem:explicit_examples_of_abstract_groups

Anonymous (anonymous@undisclosed.example.com) — 2013-08-22T16:15:48+00:00

Some explicit examples of abstract groups Problem Give examples of groups with the following properties by giving a set and defining both functions as well as the distinguished element. * A group with elements * A different group with elements, if possible

problem:exploring_the_set_products_hk_and_kh_on_d4

Anonymous (anonymous@undisclosed.example.com) — 2013-08-16T18:09:47+00:00

Exploring the set products $HK$ and $KH$ on $D_4$ Problem Consider the Cayley graph of the dihedral group shown below. Let represent the red arrow (a rotation of 90 degrees). Let represent the blue arrow (a flip). * Consider the subgropus and

problem:first_encryption_key

Anonymous (anonymous@undisclosed.example.com) — 2013-09-27T16:49:46+00:00

First Encryption Key Problem Suppose you encounter some ciphertext $(i,q,x,x,p,a,z,q)$ that you know has been encrypted using a simple shift permutation $\phi_n$ for some $n$. * Decode the ciphertext and state the original message. State the value of $n$ that was used to encode the original message.

problem:number_of_permutations

Anonymous (anonymous@undisclosed.example.com) — 2013-09-27T17:07:01+00:00

Number Of Permutations Problem Given finite set $X$ with $n$ elements, how many permutations are there of $X$? * Show that if $X=\{1\}$, then there is only one permutation of $X$. * Show that if $X=\{1,2\}$, then there are two permutations of $X$. List them.

problem:permutation_group_generated_by_s

Anonymous (anonymous@undisclosed.example.com) — 2013-08-22T15:56:21+00:00

Permutation Group Generated By $S$ Problem Let be a set. Let be a collection of permutations of . * Show that there is a permutation group that contains . * Let be the intersection of all permutation groups that contain . Show that is a permutation group.

problem:s_x_is_a_group

Anonymous (anonymous@undisclosed.example.com) — 2013-12-24T22:50:21+00:00

S X Is A Group Problem Let be any set, and let be the set of permutations on . Please show that is a group under function composition. ---------- Remarks * Make remarks with a list. ---------- $\LaTeX$ version %%%%% % DEPENDENCIES % RequiredPackages: \usepackage{tikz} % RequiredMacros: \DeclareMathOperator{\aut}{Aut} %%%%% \begin{problem} Type the problem code here. \end{problem}

problem:simple_matrix_encryption

Anonymous (anonymous@undisclosed.example.com) — 2013-08-16T17:29:02+00:00

Simple Matrix Encryption Problem Consider the matrix . Joe decides to send a message to Sam by encrypting the message with the matrix . He first takes his message and converts it to numbers by replacing A with 1, B with 2, C with 3, and so on till replacing Z with 26. He uses a 0 for spaces. After replacing the letters with numbers, he breaks the message up into chunks of 3 letters. He then multiplies each chunk of 3 by the matrix

problem:simple_shift_repetition

Anonymous (anonymous@undisclosed.example.com) — 2013-09-27T18:46:24+00:00

Simple Shift Repetition Problem Let's now devise a way to not only encrypt a message, but also keep track of who has seen the message? There are several ways to do this. Let's look at an example that involves repeated application of the same encrpytion key. For this example, let's use the encryption key $\phi_4:S\to S$ (the simple shift permutation that shifts right 4).

problem:simple_shift_repetition_game

Anonymous (anonymous@undisclosed.example.com) — 2013-11-21T11:18:29+00:00

Simple Shift Repetition Game Problem Consider the set of simple shift permutations on a 26 letter alphabet. We've shown that there are 26 different functions in this set. Consider the following game. * The first player picks an element . They then remove from

problem:soccerballahedron

Anonymous (anonymous@undisclosed.example.com) — 2013-08-14T13:50:18+00:00

Soccerballahedron Problem Construct the Cayley graph of the group defined by the group presentation $$\left< a,b\mid a^2, b^5, ababab\right>.$$ Can you think of a common object whose shape is similar to this graph? ---------- Remarks * One option ends up with a soccerball.

problem:template

Anonymous (anonymous@undisclosed.example.com) — 2013-08-15T12:42:13+00:00

@!!PAGE@ Problem Type the problem. ---------- Remarks * Make remarks with a list. ---------- $\LaTeX$ version %%%%%%%%%% % DEPENDENCIES % RequiredPackages: \usepackage{tikz} % RequiredMacros: \DeclareMathOperator{\aut}{Aut} %%%%%%%%%% \begin{problem} Type the problem code here. \end{problem}

problem:the_composition_of_permutations_is_a_permutation

Anonymous (anonymous@undisclosed.example.com) — 2013-11-21T10:30:02+00:00

The Composition Of Permutations Is A Permutation Problem Prove theorem The Composition Of Permutations Is A Permutation. As some reminders, you may use the following facts that you proved in either Math 301 or Math 340. You may use these facts without proof. * The composition of two injective functions is injective.

problem:the_game_of_scoring

Anonymous (anonymous@undisclosed.example.com) — 2013-09-19T15:00:08+00:00

The Game Of Scoring Problem The game of Scoring is a two-player game. Start by creating a pile of $n\geq 1$ objects (feel free to choose $n$ however you want). On each player's turn, they must choose 1, 2, or 3 items from the pile. Players alternate taking turns until someone takes the last object. Whoever takes the last object wins.

problem:the_game_of_scoring_misere

Anonymous (anonymous@undisclosed.example.com) — 2013-09-14T10:45:46+00:00

The Game Of Scoring Misere Problem A misere game is a game played by the regular rules with one change; whoever wins the game according the regular rules is the loser. Consider again the game of Scoring, but this time we'll play it as a misere game. * For which values of does the first player have a winning strategy when playing misere, provided each player must take 1, 2, or 3 objects?

problem:the_order_of_a_simple_shift

Anonymous (anonymous@undisclosed.example.com) — 2013-08-14T13:30:04+00:00

====== The Order of a Simple Shift ====== ==== Problem ==== Let $S = \{a,b,c,\ldots,z\}$ be the set of letters in the roman alphabet. For each $n\in\mathbb{Z}$, let $\phi_{n}:S\to S$ represent the encryption key that shifts each letter in the alphabet right $n$, where we wrap around from $z$ to $a$ for letters that need to be shifted past $z$. So $\phi_3(a)=d$, $\phi_3(b)=3$, and $\phi_3(z)=c$. - For which $n$ does $\phi_n$ not change the message? - Consider the encryption key $\phi_{9}$, w…

problem:the_set_of_simple_shift

Anonymous (anonymous@undisclosed.example.com) — 2013-09-27T16:52:57+00:00

====== The Set Of Simple Shift Permutations====== ==== Problem ==== Let $S = \{a,b,c,\ldots,z\}$ be the set of letters in the Roman alphabet. For each $n\in\mathbb{Z}$, let $\phi_{n}:S\to S$ represent the encryption key that shifts each letter in the alphabet right $n$, where we wrap around from $z$ to $a$ for letters that need to be shifted past $z$. So $\phi_3(a)=d$, $\phi_3(b)=3$, and $\phi_3(z)=c$. We've called this a simple shift permutation of $S$. - For which $n$ does $\phi_n$ not chan…

problem:the_set_of_simple_shift_permutations

Anonymous (anonymous@undisclosed.example.com) — 2013-09-27T16:53:48+00:00

The Set Of Simple Shift Permutations Problem Let $S = \{a,b,c,\ldots,z\}$ be the set of letters in the Roman alphabet. For each $n\in\mathbb{Z}$, let $\phi_{n}:S\to S$ represent the encryption key that shifts each letter in the alphabet right $n$, where we wrap around from $z$ to $a$ for letters that need to be shifted past $z$. So $\phi_3(a)=d$, $\phi_3(b)=3$, and $\phi_3(z)=c$. We've called this a simple shift permutation of $S$.

problem:the_span_of_a_simple_shift

Anonymous (anonymous@undisclosed.example.com) — 2013-11-21T14:55:05+00:00

The Span Of A Simple Shift Problem Suppose that our alphabet consists of only 12 letters . Let be the set of simple shift permutations on this 12 letter alphabet (we wrap around from to ). * For each , list the elements in . * For which does

problem:when_do_two_simple_shifts_span_the_same_set

Anonymous (anonymous@undisclosed.example.com) — 2013-11-25T13:46:37+00:00

====== When Do Two Simple Shifts Span The Sameset ====== ==== Problem ==== Consider the sets $H_{12}$ and $H_{15}$ of simple shift permutations on alphabets with 12 and 15 letters respectively. - For each $k\in\{0,1,2,\ldots,11\}$, make a list of the elements in $H_{12}$ that are in $\text{span}(\{\phi_k\})$ and state the order of $\phi_k$ as an element of $H_{12}$. - For each $k\in\{0,1,2,\ldots,14\}$, make a list of the elements in $H_{15}$ that are in $\text{span}(\{\phi_k\})$ and state …

problem:why_we_need_associative

Anonymous (anonymous@undisclosed.example.com) — 2013-08-09T18:16:01+00:00

====== Why we need associative ====== FIXME Add in the axioms of multiplication, as well as a few basic rules about solving equations (such as if $x=y$, then $ax=ay$. ==== Problem ==== Consider the equation $2x=3$. Carefully solve this problem for $x$, where at each stage of your computation you should explain what rule you are using to performing. Do not skip any steps in your computation. Did you use the commutative rule of multiplication above? Look through your work above for the co…