# Learning Objectives for MTH 410: Modern Algebra 2 This document can also be found online: * At the course Blackboard site under `Course Documents > MTH 410 Learning Objectives` * On Github Gists at: [https://gist.github.com/RobertTalbert/c79a2c5de23197434f49](https://gist.github.com/RobertTalbert/c79a2c5de23197434f49) Legend: * CC = Learning objectives to be assessed through Concept Checks. * M = Learning objectives to be assessed through Learning Modules. * (CORE) = Learning objectives designated as belonging to the 20 CORE-M learning objectives for Modules, assessed during timed assessment periods. To "instantiate" a definition means to construct or state examples of that definition, and to create a non-example of that definition. # MTH 410 Learning Objectives in order of appearance ## 19: Symmetry * CC.1: State and instantiate the definitions of the following terms: Rigid motion; symmetry * (CORE) M.1: Create or complete an operation table for symmetries of a figure. ## 20: An Introduction to Groups * CC.2: State and instantiate the definitions of the following terms: Group; Abelian group; finite/infinite order of a group; finite/infinite group; unit * CC.3: State the following mathematical results: Theorem 20.7 * (CORE) M.2: Determine whether a set with a given binary operation is or is not a group. * (CORE) M.3: Given a group (especially the standard examples of groups on page 285), do the following: identify the identity element; identify the inverse of a given element; perform operations in the group; determine whether it is Abelian; determine its order; and describe its group of units. ## 21: Integer Powers of Elements of a Group * CC.4: State and instantiate the definitions of the following terms: Power of an element in a group * CC.5: State the following mathematical results: Theorem 21.4; Theorem 21.5 * (CORE) M.4: Use properties of exponentiation to manipulate powers of group elements. ## 22: Subgroups * CC.6: State and instantiate the definitions of the following terms: Subgroup; center of a group; cyclic subgroup generated by an element; cyclic group; order of an element in a group * CC.7: State the following mathematical results: Theorem 22.4 (Subgroup Test) * (CORE) M.5: Determine whether a subset of a group is a subgroup of that group by applying the Subgroup Test. * (CORE) M.6: Determine the subgroup of a group that is generated by a single element <span class="math">_a_</span> (that is, the cyclic subgroup generated by <span class="math">_a_</span>). ## 23: Subgroups of Cyclic Groups * CC.8: State the following mathematical results: Theorem 23.2; Theorem 23.5; Theorem 23.6; Theorem 23.7; Theorem 23.10 * M.7: Use Theorem 23.2 to determine information about subgroups of a cyclic group. * (CORE) M.8: Use Theorem 23.5 to determine information about the order of an element in a cyclic group. * (CORE) M.9: Determine subgroups of a given order in a cyclic group using Theorem 23.7. ## 24: The Dihedral Groups * CC.9: State and instantiate the definitions of the following terms: Dihedral group of order 2n; a subset that generates a group * M.10: Determine a presentation for a dihedral group. ## 25: The Symmetric Groups * CC.10: State and instantiate the definitions of the following terms: ermutation; symmetric group of degree n; disjoint cycles; even and odd permutations; alternating group <span class="math">_A_<sub>_n_</sub></span> * CC.11: State the following mathematical results: Theorem 25.4; Theorem 25.11; Theorem 25.14 * CC.12: Write a permutation in cycle notation, and decompose a permutation into a product of disjoint cycles. * M.11: Determine whether a permutation is even or odd. ## 26: Cosets and Lagrange's Theorem * CC.13: State and instantiate the definitions of the following terms: The relation defined in Definition 26.3; left and right cosets of a subgroup; index of a subgroup * CC.14: State the following mathematical results: Theorem 26.7; Theorem 26.11 (Lagrange's Theorem); Corollary 26.13; Corollary 26.14 * (CORE) M.12: Given a group G, a subgroup H, and an group element a, determine the left coset aH of H in G and the right coset Ha of H in G. * (CORE) M.13: Use Lagrange's Theorem to determine information about the order of a subgroup of a group. * (CORE) M.14: Use the Corollaries to Lagrange's Theorem to determine information about subgroups of a group and about powers of elements of a group. ## 27: Normal Subgroups and Quotient Groups * CC.15: State and instantiate the definitions of the following terms: The set G/H; normal subgroup; quotient group G/N; simple group * CC.16: State the following mathematical results: Theorem 27.5; Theorem 27.10; Theorem 27.11 (Cauchy's Theorem for Finite Abelian Groups) * (CORE) M.15: Determine the set G/H of distinct left cosets of H in G. * (CORE) M.16: Determine whether a subgroup of a group is normal. * (CORE) M.17: Determine the quotient group G/N of G by a normal subgroup N. ## 28: Products of Groups * CC.17: State and instantiate the definitions of the following terms: External direct product of two groups; internal direct product of two groups * CC.18: State the following mathematical results: Theorem 28.5; Theorem 28.6; Theorem 28.9; Theorem 28.15 * (CORE) M.18: Determine the external direct product of two groups. * M.19: Determine the order of an element in a direct product. ## 29: Group Isomorphisms and Invariants * CC.19; State and instantiate the definitions of the following terms: Isomorphism of groups; well-defined mapping * CC.20: State the following mathematical results: Theorem at the end of Activity 29.9; Theorem 29.14; Theorem 29.16; Theorem 29.18; Theorem 29.19; Theorem 29.24; Corollary 29.25 * (CORE) M.20: Determine whether a function between two groups is an isomorphism. * (CORE) M.21: Show two groups are isomorphic by constructing an isomorphism between them. * M.22: Determine whether a mapping is well-defined. * (CORE) M.23: Show two groups are non-isomorphic by identifying differences on one or more invariants. ## 30: Homomorphisms and Isomorphism Theorems * CC.21: State and instantiate the definitions of the following terms: Homomorphism of groups; epimorphism; monomorphism; homomorphic image; kernel of a homomorphism; image of a homomorphism * CC.22: State the following mathematical results: heorem 30.4; Theorem 30.8; Theorem 30.13 (The First Isomorphism Theorem); Theorem 30.16 (The Second Isomorphism Theorem); Theorem 30.18 (The Third Isomorphism Theorem); Theorem 30.19 (The Fourth Isomorphism Theorem); Theorem 30.22 * (CORE) M.24: Determine whether a function between two groups is a homomorphism (epimorphism, monomorphism). * (CORE) M.25: Determine the kernel and image of a homomorphism. ## 31: The Fundamental Theorem of Finite Abelian Groups * CC.23: State and instantiate the definitions of the following terms: p-group; p-primary component of a group; * CC.24: State the following mathematical results: Theorem 31.3; Corollary 31.6; Theorem 31.13 (Fundamental Theorem of Finite Abelian Groups) ## 32: The First Sylow Theorem * CC.25: State and instantiate the definitions of the following terms: The "conjugate" relation; conjugacy class; centralizer; Sylow p-subgroup * CC.26: State the following mathematical results: Theorem 32.12 (The Class Equation); Theorem 32.14; Corollary 32.16; Theorem 32.17 (Cauchy's Theorem); Theorem 32.20 (First Sylow Theorem) * M.26: Determine if two elements in a group are conjugate. * M.27: Determine the conjugacy class of a group element. * M.28: Determine the centralizer of a group. ## 33: The Second and Third Sylow Theorem * CC.27: State and instantiate the definitions of the following terms: Normalizer * CC.28: State the following mathematical results: Lemma 33.3; Lemma 33.9; Lemma 33.10; Theorem 33.12 (Second Sylow Theorem); Theorem 33.13 (Third Sylow Theorem) * M.29: Determine the normalizer of a subgroup in a group. ## 16: Rings: Ideals and Homomorphisms Note: This investigation will occupy several class meetings. * CC.29: State and instantiate the definitions of the following terms: Ideal; principal ideal; principal ideal domain; Euclidean domain; associates; congruence modulo an ideal; maximal ideal; prime ideal; homomorphism of rings; monomorphism, epimorphism, and isomorphism of rings; kernel of a ring homomorphism; image of a ring homomorphism * CC.30: State the following mathematical results: Theorem 16.4 (Ideal Test); Theorem 16.8; Theorem 16.10; Lemma 16.11; Theorem 16.13; Lemma 16.16; Theorem 16.24; Theorem 16.27; Euclid's Lemma; Theorem 16.34; Theorem 16.39; Theorem 16.42; Theorem 16.43; Theorem 16.46; Theorem 16.48 (First Isomorphism Theorem for Rings) * (CORE) M.30: Determine whether a subset of a ring is an ideal of that ring. * (CORE) M.31: Determine whether an ideal of a ring is a principal ideal; and given a principal ideal, determine its elements and a generator. * M.32: Determine if a ring is a principal ideal domain. * M.33: Determine if a ring is a Euclidean domain. * M.34: Given a ring R and an ideal I, determine whether two ring elements are congruent modulo I. * M.35: Determine if an ideal in a ring is maximal or prime. * (CORE) M.36: Determine if a function between two rings is a ring homomorphism. * (CORE) M.37: Determine the kernel and image of a ring homomorphism. # MTH 410 Learning Objectives by type ## Concept Check (CC) Objectives * CC.1: State and instantiate the definitions of the following terms: Rigid motion; symmetry * CC.2: State and instantiate the definitions of the following terms: Group; Abelian group; finite/infinite order of a group; finite/infinite group; unit * CC.3: State the following mathematical results: Theorem 20.7 * CC.4: State and instantiate the definitions of the following terms: Power of an element in a group * CC.5: State the following mathematical results: Theorem 21.4; Theorem 21.5 * CC.6: State and instantiate the definitions of the following terms: Subgroup; center of a group; cyclic subgroup generated by an element; cyclic group; order of an element in a group * CC.7: State the following mathematical results: Theorem 22.4 (Subgroup Test) * CC.8: State the following mathematical results: Theorem 23.2; Theorem 23.5; Theorem 23.6; Theorem 23.7; Theorem 23.10 * CC.9: State and instantiate the definitions of the following terms: Dihedral group of order 2n; a subset that generates a group * CC.10: State and instantiate the definitions of the following terms: ermutation; symmetric group of degree n; disjoint cycles; even and odd permutations; alternating group <span class="math">_A_<sub>_n_</sub></span> * CC.11: State the following mathematical results: Theorem 25.4; Theorem 25.11; Theorem 25.14 * CC.12: Write a permutation in cycle notation, and decompose a permutation into a product of disjoint cycles. * CC.13: State and instantiate the definitions of the following terms: The relation defined in Definition 26.3; left and right cosets of a subgroup; index of a subgroup * CC.14: State the following mathematical results: Theorem 26.7; Theorem 26.11 (Lagrange's Theorem); Corollary 26.13; Corollary 26.14 * CC.15: State and instantiate the definitions of the following terms: The set G/H; normal subgroup; quotient group G/N; simple group * CC.16: State the following mathematical results: Theorem 27.5; Theorem 27.10; Theorem 27.11 (Cauchy's Theorem for Finite Abelian Groups) * CC.17: State and instantiate the definitions of the following terms: External direct product of two groups; internal direct product of two groups * CC.18: State the following mathematical results: Theorem 28.5; Theorem 28.6; Theorem 28.9; Theorem 28.15 * CC.19; State and instantiate the definitions of the following terms: Isomorphism of groups; well-defined mapping * CC.20: State the following mathematical results: Theorem at the end of Activity 29.9; Theorem 29.14; Theorem 29.16; Theorem 29.18; Theorem 29.19; Theorem 29.24; Corollary 29.25 * CC.21: State and instantiate the definitions of the following terms: Homomorphism of groups; epimorphism; monomorphism; homomorphic image; kernel of a homomorphism; image of a homomorphism * CC.22: State the following mathematical results: heorem 30.4; Theorem 30.8; Theorem 30.13 (The First Isomorphism Theorem); Theorem 30.16 (The Second Isomorphism Theorem); Theorem 30.18 (The Third Isomorphism Theorem); Theorem 30.19 (The Fourth Isomorphism Theorem); Theorem 30.22 * CC.23: State and instantiate the definitions of the following terms: p-group; p-primary component of a group; * CC.24: State the following mathematical results: Theorem 31.3; Corollary 31.6; Theorem 31.13 (Fundamental Theorem of Finite Abelian Groups) * CC.25: State and instantiate the definitions of the following terms: The "conjugate" relation; conjugacy class; centralizer; Sylow p-subgroup * CC.26: State the following mathematical results: Theorem 32.12 (The Class Equation); Theorem 32.14; Corollary 32.16; Theorem 32.17 (Cauchy's Theorem); Theorem 32.20 (First Sylow Theorem) * CC.27: State and instantiate the definitions of the following terms: Normalizer * CC.28: State the following mathematical results: Lemma 33.3; Lemma 33.9; Lemma 33.10; Theorem 33.12 (Second Sylow Theorem); Theorem 33.13 (Third Sylow Theorem) * CC.29: State and instantiate the definitions of the following terms: Ideal; principal ideal; principal ideal domain; Euclidean domain; associates; congruence modulo an ideal; maximal ideal; prime ideal; homomorphism of rings; monomorphism, epimorphism, and isomorphism of rings; kernel of a ring homomorphism; image of a ring homomorphism * CC.30: State the following mathematical results: Theorem 16.4 (Ideal Test); Theorem 16.8; Theorem 16.10; Lemma 16.11; Theorem 16.13; Lemma 16.16; Theorem 16.24; Theorem 16.27; Euclid's Lemma; Theorem 16.34; Theorem 16.39; Theorem 16.42; Theorem 16.43; Theorem 16.46; Theorem 16.48 (First Isomorphism Theorem for Rings) ## Module (M) Objectives * M.1: Create or complete an operation table for symmetries of a figure. * M.2: Determine whether a set with a given binary operation is or is not a group. * M.3: Given a group (especially the standard examples of groups on page 285), do the following: identify the identity element; identify the inverse of a given element; perform operations in the group; determine whether it is Abelian; determine its order; and describe its group of units. * M.4: Use properties of exponentiation to manipulate powers of group elements. * M.5: Determine whether a subset of a group is a subgroup of that group by applying the Subgroup Test. * M.6: Determine the subgroup of a group that is generated by a single element <span class="math">_a_</span> (that is, the cyclic subgroup generated by <span class="math">_a_</span>). * M.7: Use Theorem 23.2 to determine information about subgroups of a cyclic group. * M.8: Use Theorem 23.5 to determine information about the order of an element in a cyclic group. * M.9: Determine subgroups of a given order in a cyclic group using Theorem 23.7. * M.10: Determine a presentation for a dihedral group. * M.11: Determine whether a permutation is even or odd. * M.12: Given a group G, a subgroup H, and an group element a, determine the left coset aH of H in G and the right coset Ha of H in G. * M.13: Use Lagrange's Theorem to determine information about the order of a subgroup of a group. * M.14: Use the Corollaries to Lagrange's Theorem to determine information about subgroups of a group and about powers of elements of a group. * M.15: Determine the set G/H of distinct left cosets of H in G. * M.16: Determine whether a subgroup of a group is normal. * M.17: Determine the quotient group G/N of G by a normal subgroup N. * M.18: Determine the external direct product of two groups. * M.19: Determine the order of an element in a direct product. * M.20: Determine whether a function between two groups is an isomorphism. * M.21: Show two groups are isomorphic by constructing an isomorphism between them. * M.22: Determine whether a mapping is well-defined. * M.23: Show two groups are non-isomorphic by identifying differences on one or more invariants. * M.24: Determine whether a function between two groups is a homomorphism (epimorphism, monomorphism). * M.25: Determine the kernel and image of a homomorphism. * M.26: Determine if two elements in a group are conjugate. * M.27: Determine the conjugacy class of a group element. * M.28: Determine the centralizer of a group. * M.29: Determine the normalizer of a subgroup in a group. * M.30: Determine whether a subset of a ring is an ideal of that ring. * M.31: Determine whether an ideal of a ring is a principal ideal; and given a principal ideal, determine its elements and a generator. * M.32: Determine if a ring is a principal ideal domain. * M.33: Determine if a ring is a Euclidean domain. * M.34: Given a ring R and an ideal I, determine whether two ring elements are congruent modulo I. * M.35: Determine if an ideal in a ring is maximal or prime. * M.36: Determine if a function between two rings is a ring homomorphism. * M.37: Determine the kernel and image of a ring homomorphism. ## Core Module (CORE-M) Objectives These are a subset of the M Objectives above. They represent the top -- objectives in the course, so important that you will be assessed on them in both timed and untimed settings. There are 24 of these in all. * M.1: Create or complete an operation table for symmetries of a figure. * M.2: Determine whether a set with a given binary operation is or is not a group. * M.3: Given a group (especially the standard examples of groups on page 285), do the following: identify the identity element; identify the inverse of a given element; perform operations in the group; determine whether it is Abelian; determine its order; and describe its group of units. * M.4: Use properties of exponentiation to manipulate powers of group elements. * M.5: Determine whether a subset of a group is a subgroup of that group by applying the Subgroup Test. * M.6: Determine the subgroup of a group that is generated by a single element <span class="math">_a_</span> (that is, the cyclic subgroup generated by <span class="math">_a_</span>). * M.8: Use Theorem 23.5 to determine information about the order of an element in a cyclic group. * M.9: Determine subgroups of a given order in a cyclic group using Theorem 23.7. * M.12: Given a group G, a subgroup H, and an group element a, determine the left coset aH of H in G and the right coset Ha of H in G. * M.13: Use Lagrange's Theorem to determine information about the order of a subgroup of a group. * M.14: Use the Corollaries to Lagrange's Theorem to determine information about subgroups of a group and about powers of elements of a group. * M.15: Determine the set G/H of distinct left cosets of H in G. * M.16: Determine whether a subgroup of a group is normal. * M.17: Determine the quotient group G/N of G by a normal subgroup N. * M.18: Determine the external direct product of two groups. * M.20: Determine whether a function between two groups is an isomorphism. * M.21: Show two groups are isomorphic by constructing an isomorphism between them. * M.23: Show two groups are non-isomorphic by identifying differences on one or more invariants. * M.24: Determine whether a function between two groups is a homomorphism (epimorphism, monomorphism). * M.25: Determine the kernel and image of a homomorphism. * M.30: Determine whether a subset of a ring is an ideal of that ring. * M.31: Determine whether an ideal of a ring is a principal ideal; and given a principal ideal, determine its elements and a generator. * M.36: Determine if a function between two rings is a ring homomorphism. * M.37: Determine the kernel and image of a ring homomorphism.