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- https://math.libretexts.org/Bookshelves/Abstract_and_Geometric_Algebra/An_Inquiry-Based_Approach_to_Abstract_Algebra_(Ernst)/08%3A_An_Introduction_to_Rings
- https://math.libretexts.org/Bookshelves/Abstract_and_Geometric_Algebra/An_Inquiry-Based_Approach_to_Abstract_Algebra_(Ernst)/04%3A_Families_of_Groups/4.01%3A_Cyclic_GroupsRecall that if \(G\) is a group and \(g\in G\), then the cyclic subgroup generated by \(g\) is given by \[\langle g\rangle =\{g^k\mid k\in\mathbb{Z}\}.\] It is important to point out that \(\langle g\...Recall that if \(G\) is a group and \(g\in G\), then the cyclic subgroup generated by \(g\) is given by \[\langle g\rangle =\{g^k\mid k\in\mathbb{Z}\}.\] It is important to point out that \(\langle g\rangle\) may be finite or infinite. If \(G\) is a group and \(g\in G\) such that \(\langle g\rangle\) is a finite group, then the order of \(g\) is the smallest positive integer \(n\) such that \(g^n=e\).
- https://math.libretexts.org/Bookshelves/Abstract_and_Geometric_Algebra/An_Inquiry-Based_Approach_to_Abstract_Algebra_(Ernst)This free and open-source textbook is designed to used with an inquiry-based learning (IBL) approach to a first-semester undergraduate abstract algebra course. The textbook starts with groups (up to t...This free and open-source textbook is designed to used with an inquiry-based learning (IBL) approach to a first-semester undergraduate abstract algebra course. The textbook starts with groups (up to the First Isomorphism Theorem) and finishes with an introduction to rings (up to quotients by maximal and prime ideals). While the textbook covers many of the standard topics, the focus is on building intuition and emphasizes visualization.
- https://math.libretexts.org/Bookshelves/Abstract_and_Geometric_Algebra/An_Inquiry-Based_Approach_to_Abstract_Algebra_(Ernst)/05%3A_Cosets_Lagranges_Theorem_and_Normal_Subgroups
- https://math.libretexts.org/Bookshelves/Abstract_and_Geometric_Algebra/An_Inquiry-Based_Approach_to_Abstract_Algebra_(Ernst)/06%3A_Products_and_Quotients_of_Groups/6.01%3A_Products_of_GroupsEvery finitely generated abelian group \(G\) is isomorphic to a direct product of cyclic groups of the form \[\mathbb{Z}_{p_1^{n_1}}\times \mathbb{Z}_{p_2^{n_2}}\times \cdots \times \mathbb{Z}_{p_r^{n...Every finitely generated abelian group \(G\) is isomorphic to a direct product of cyclic groups of the form \[\mathbb{Z}_{p_1^{n_1}}\times \mathbb{Z}_{p_2^{n_2}}\times \cdots \times \mathbb{Z}_{p_r^{n_r}}\times \mathbb{Z}^k,\] where each \(p_i\) is a prime number (not necessarily distinct).
- https://math.libretexts.org/Bookshelves/Abstract_and_Geometric_Algebra/An_Inquiry-Based_Approach_to_Abstract_Algebra_(Ernst)/02%3A_Introduction_to_Groups/2.05%3A_Group_TablesRecall that we could represent a binary operation on a finite set using a table. Since groups have binary operations at their core, we can represent a finite group (i.e., a group with finitely many el...Recall that we could represent a binary operation on a finite set using a table. Since groups have binary operations at their core, we can represent a finite group (i.e., a group with finitely many elements) using a table, called a group table.
- https://math.libretexts.org/Bookshelves/Abstract_and_Geometric_Algebra/An_Inquiry-Based_Approach_to_Abstract_Algebra_(Ernst)/01%3A_Introduction/1.02%3A_What_Should_You_ExpectUp to this point, it is possible that your experience of mathematics has been about using formulas and algorithms. You are used to being asked to do things like: “solve for x", “take the derivative of...Up to this point, it is possible that your experience of mathematics has been about using formulas and algorithms. You are used to being asked to do things like: “solve for x", “take the derivative of this function", “integrate this function", etc. Your progress will be fueled by your ability to wrestle with mathematical ideas and to prove theorems. As you work through the book, you will find that you have ideas for proofs, but you are unsure of them.
- https://math.libretexts.org/Bookshelves/Abstract_and_Geometric_Algebra/An_Inquiry-Based_Approach_to_Abstract_Algebra_(Ernst)/03%3A_Subgroups_and_Isomorphisms/3.01%3A_SubgroupsIf you ignore the labels on the vertices and just pay attention to the configuration of arrows, it appears that there are two copies of the Cayley diagram for \(R_4\) in the Cayley diagram for \(D_4\)...If you ignore the labels on the vertices and just pay attention to the configuration of arrows, it appears that there are two copies of the Cayley diagram for \(R_4\) in the Cayley diagram for \(D_4\). This means that if we want to prove that a certain subset \(H\) is a subgroup of a group \(G\), then one of the things we must do is verify that \(H\) is in fact nonempty.
- https://math.libretexts.org/Bookshelves/Abstract_and_Geometric_Algebra/An_Inquiry-Based_Approach_to_Abstract_Algebra_(Ernst)/02%3A_Introduction_to_Groups/2.01%3A_A_First_ExampleIt’s not too difficult to prove—but we will omit the details—that we can generate \(\text{Spin}_{3\times 3}\) with the following subset of 9 spins: \[T=\{s_{11}, s_{12}, s_{23}, s_{36}, s_{56}, s_{45}...It’s not too difficult to prove—but we will omit the details—that we can generate \(\text{Spin}_{3\times 3}\) with the following subset of 9 spins: \[T=\{s_{11}, s_{12}, s_{23}, s_{36}, s_{56}, s_{45}, s_{47}, s_{78}, s_{89}\}.\] That is, every net action in \(\text{Spin}_{3\times 3}\) corresponds to a word consisting of the spins from \(T\).
- https://math.libretexts.org/Bookshelves/Abstract_and_Geometric_Algebra/An_Inquiry-Based_Approach_to_Abstract_Algebra_(Ernst)/09%3A_Appendix/9.03%3A_A.3-_Definitions_in_MathematicsLikewise, we often hear calculus students speak of a continuous function as one whose graph can be drawn “without picking up the pencil.” This definition is descriptive. (As we learned in calculus the...Likewise, we often hear calculus students speak of a continuous function as one whose graph can be drawn “without picking up the pencil.” This definition is descriptive. (As we learned in calculus the picking-up-the-pencil description is not a perfect description of continuous functions.) This is not a mathematical definition.
- https://math.libretexts.org/Bookshelves/Abstract_and_Geometric_Algebra/An_Inquiry-Based_Approach_to_Abstract_Algebra_(Ernst)/07%3A_Homomorphisms_and_the_Isomorphism_Theorems/7.02%3A_The_Isomorphism_TheoremsThe next theorem is a generalization of Theorem [thm:orderImage] and follows from the First Isomorphism Theorem together with Lagrange’s Theorem. In particular, every subgroup \(G\) is of the form \(H...The next theorem is a generalization of Theorem [thm:orderImage] and follows from the First Isomorphism Theorem together with Lagrange’s Theorem. In particular, every subgroup \(G\) is of the form \(H/N\) for some subgroup \(H\) of \(G\) containing \(N\) (namely, its preimage in \(G\) under the canonical projection homomorphism from \(G\) to \(G/N\).) This bijection has the following properties: for all \(H,K \leq G\) with \(N\leq H\) and \(N\subseteq K\), we have
