Learning Objectives for MTH 410: Modern Algebra 2

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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 a (that is, the cyclic subgroup generated by a).

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 A_n
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 A_n
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 a (that is, the cyclic subgroup generated by a).
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 a (that is, the cyclic subgroup generated by a).
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.