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3: Groups II
https://math.libretexts.org/Bookshelves/Abstract_and_Geometric_Algebra/Introduction_to_Algebraic_Structures_(Denton)/03%3A_Groups_II
In this chapter we explore the structure of groups using Cayley graphs and generating sets. We also learn about Lagrange's theorem, which gives an interesting numerical relationship between the size o...In this chapter we explore the structure of groups using Cayley graphs and generating sets. We also learn about Lagrange's theorem, which gives an interesting numerical relationship between the size of a group and the size of a subgroup. Contributors and Attributions Tom Denton (Fields Institute/York University in Toronto)
7.1: Juggling With Two Operations
https://math.libretexts.org/Bookshelves/Abstract_and_Geometric_Algebra/Introduction_to_Algebraic_Structures_(Denton)/07%3A_Rings_I/7.01%3A_Juggling_With_Two_Operations
There are many reasons to study ring theory, often having to do with generalizing the properties that we observe in many of the rings we deal with in day-to-day life, like the integers and the rationa...There are many reasons to study ring theory, often having to do with generalizing the properties that we observe in many of the rings we deal with in day-to-day life, like the integers and the rational numbers. The basic idea of algebraic geometry is to study geometry using zeroes of polynomials: for example, a line in the plane can be thought of as the zeroes of the polynomial $f(x,y) = y-mx-b$ where $m$ and $b$ are constants.
4.1: Homomorphisms
https://math.libretexts.org/Bookshelves/Abstract_and_Geometric_Algebra/Introduction_to_Algebraic_Structures_(Denton)/04%3A_Groups_III/4.01%3A_Homomorphisms
But not just any functions: the symmetries don't distort the triangle in any way: For example, The center of the triangle never gets moved closer to one of the vertices. Then to show that $\phi$ is ...But not just any functions: the symmetries don't distort the triangle in any way: For example, The center of the triangle never gets moved closer to one of the vertices. Then to show that $\phi$ is a homomorphism, we need to check that $\phi(n+m)=\phi(n)+\phi(m)$ ; the operation before applying $\phi$ is the same as the operation after applying $\phi$ . Show that $rho$ is a isomorphism. (Hint: Show that the map is a homomorphism, and argue that the two sets have the same cardinality.)
2.4: Permutations
https://math.libretexts.org/Bookshelves/Abstract_and_Geometric_Algebra/Introduction_to_Algebraic_Structures_(Denton)/02%3A_Groups_I/2.04%3A_Permutations
In this sense, the permutation is a special kind of function from the set of objects back to itself. (By special, I mean it's a bijection, which is to say a one-to-one and onto function.) (TODO: Wikip...In this sense, the permutation is a special kind of function from the set of objects back to itself. (By special, I mean it's a bijection, which is to say a one-to-one and onto function.) (TODO: Wikipedia link) A permutation of these objects is then the list $[\sigma(a), \sigma(b), \sigma(c)]$ ; this list is called the one-line notation for $\sigma$ .
InfoPage
https://math.libretexts.org/Bookshelves/Abstract_and_Geometric_Algebra/Introduction_to_Algebraic_Structures_(Denton)/00%3A_Front_Matter/02%3A_InfoPage
The LibreTexts libraries are Powered by MindTouch ® and are supported by the Department of Education Open Textbook Pilot Project, the UC Davis Office of the Provost, the UC Davis Library, the Californ...The LibreTexts libraries are Powered by MindTouch ® and are supported by the Department of Education Open Textbook Pilot Project, the UC Davis Office of the Provost, the UC Davis Library, the California State University Affordable Learning Solutions Program, and Merlot.
Front Matter
https://math.libretexts.org/Bookshelves/Abstract_and_Geometric_Algebra/Introduction_to_Algebraic_Structures_(Denton)/00%3A_Front_Matter
Introduction to Algebraic Structures (Denton)
https://math.libretexts.org/Bookshelves/Abstract_and_Geometric_Algebra/Introduction_to_Algebraic_Structures_(Denton)
An algebraic structure is a set (called carrier set or underlying set) with one or more finitary operations defined on it that satisfies a list of axioms. Examples of algebraic structures include grou...An algebraic structure is a set (called carrier set or underlying set) with one or more finitary operations defined on it that satisfies a list of axioms. Examples of algebraic structures include groups, rings, fields, and lattices.
8: Rings II
https://math.libretexts.org/Bookshelves/Abstract_and_Geometric_Algebra/Introduction_to_Algebraic_Structures_(Denton)/08%3A_Rings_II
In working with different rings (and different kinds of rings) questions quickly arise about which familiar properties of one ring might carry over to another ring. To illustrate this kind of question...In working with different rings (and different kinds of rings) questions quickly arise about which familiar properties of one ring might carry over to another ring. To illustrate this kind of question, we'll spend this chapter talking about division. Contributors and Attributions Tom Denton (Fields Institute/York University in Toronto)
6.3: Counting
https://math.libretexts.org/Bookshelves/Abstract_and_Geometric_Algebra/Introduction_to_Algebraic_Structures_(Denton)/06%3A_Group_Actions/6.03%3A_Counting
Then the group $G$ might be symmetries of the cube, and the set $S$ would be all ways of painting the cube in fixed position: What we're really trying to count is $|S/\mathord G|$ . First notice ...Then the group $G$ might be symmetries of the cube, and the set $S$ would be all ways of painting the cube in fixed position: What we're really trying to count is $|S/\mathord G|$ . First notice that the sum of the size of the fixed sets $S^g$ is equal to the sum of the size of the stabilizer groups $G_s$ : Both are counting the number of pairs $(g,s)$ such that $g\cdot s=s$ .
9.2: Linear Independence
https://math.libretexts.org/Bookshelves/Abstract_and_Geometric_Algebra/Introduction_to_Algebraic_Structures_(Denton)/09%3A_Vector_Spaces/9.02%3A_Linear_Independence
Let $\mathbb{R}^\infty$ be the vector space of sequences of elements of $\mathbb{R}$ . (ie, the space of sequences $r=(r_1,r_2, r_3, \ldots)$ , with coordinate-wise addition and the usual scalar m...Let $\mathbb{R}^\infty$ be the vector space of sequences of elements of $\mathbb{R}$ . (ie, the space of sequences $r=(r_1,r_2, r_3, \ldots)$ , with coordinate-wise addition and the usual scalar multiplication.) Let $r_i\in \mathbb{R}^\infty$ be the sequence with $(e_i)_i=1$ and $(e_i)_j=0$ for all $j\neq i$ . You might note that the sum of all of the elements in $S$ (with all coefficients in the sum equal to $1$ ) seems to be the $0$ -vector.
Back Matter
https://math.libretexts.org/Bookshelves/Abstract_and_Geometric_Algebra/Introduction_to_Algebraic_Structures_(Denton)/zz%3A_Back_Matter

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