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- https://math.libretexts.org/Bookshelves/Abstract_and_Geometric_Algebra/First-Semester_Abstract_Algebra%3A_A_Structural_Approach_(Sklar)/07%3A_The_Wonderful_World_of_CosetsBut if H is a subgroup of a group G, if we only study H we lose all the information about G's structure “outside” of H. We might hope that G−H (that is, the set of elements of ...But if H is a subgroup of a group G, if we only study H we lose all the information about G's structure “outside” of H. We might hope that G−H (that is, the set of elements of G that are not in H) is also a subgroup of G, but we immediately see that cannot be the case since the identity element of G must be in H, and H∩(G−H)=∅.
- https://math.libretexts.org/Bookshelves/Abstract_and_Geometric_Algebra/First-Semester_Abstract_Algebra%3A_A_Structural_Approach_(Sklar)/05%3A_Cyclic_Groups/5.02%3A_The_Subgroup_Lattices_of_Cyclic_GroupsWe now explore the subgroups of cyclic groups.
- https://math.libretexts.org/Bookshelves/Abstract_and_Geometric_Algebra/First-Semester_Abstract_Algebra%3A_A_Structural_Approach_(Sklar)/09%3A_The_Isomorphism_Theorem/9.02%3A_The_Second_and_Third_Isomorphism_TheoremsThe following theorems can be proven using the First Isomorphism Theorem. They are very useful in special cases.
- https://math.libretexts.org/Bookshelves/Abstract_and_Geometric_Algebra/First-Semester_Abstract_Algebra%3A_A_Structural_Approach_(Sklar)/05%3A_Cyclic_Groups
- https://math.libretexts.org/Bookshelves/Abstract_and_Geometric_Algebra/First-Semester_Abstract_Algebra%3A_A_Structural_Approach_(Sklar)/08%3A_Factor_Groups/8.01%3A_MotivationWe mentioned previously that given a subgroup H of G, we'd like to use H to get at some understanding of G's entire structure. Recall that we've defined G/H to be the set of all left cosets of...We mentioned previously that given a subgroup H of G, we'd like to use H to get at some understanding of G's entire structure. Recall that we've defined G/H to be the set of all left cosets of H in G. What we'd like to do now is equip G/H with some operation under which G/H is a group!
- https://math.libretexts.org/Bookshelves/Abstract_and_Geometric_Algebra/First-Semester_Abstract_Algebra%3A_A_Structural_Approach_(Sklar)/02%3A_Groups/2.02%3A_Exercises%2C_Part_IThis page contains part I of the exercises for Chapter 2.
- https://math.libretexts.org/Bookshelves/Abstract_and_Geometric_Algebra/First-Semester_Abstract_Algebra%3A_A_Structural_Approach_(Sklar)/09%3A_The_Isomorphism_Theorem/9.03%3A_ExercisesThis page contains the exercises for Chapter 9.
- https://math.libretexts.org/Bookshelves/Abstract_and_Geometric_Algebra/First-Semester_Abstract_Algebra%3A_A_Structural_Approach_(Sklar)/06%3A_Permutation_and_Dihedral_Groups/6.06%3A_ExercisesThis page contains the exercises for Chapter 6.
- https://math.libretexts.org/Bookshelves/Abstract_and_Geometric_Algebra/First-Semester_Abstract_Algebra%3A_A_Structural_Approach_(Sklar)/03%3A_Homomorphisms_and_Isomorphisms/3.02%3A_Definitions_of_Homomorphisms_and_IsomorphismsIntuitively, you can think of a homomorphism as a “structure-preserving” map: if you multiply and then apply homormorphism, you get the same result as when you first apply homomorphism and then multip...Intuitively, you can think of a homomorphism as a “structure-preserving” map: if you multiply and then apply homormorphism, you get the same result as when you first apply homomorphism and then multiply. Isomorphisms, then, are both structure-preserving and cardinality-preserving. Homomorphisms from a group G to itself are called endomorphisms, and isomorphisms from a group to itself are called automorphisms.
- https://math.libretexts.org/Bookshelves/Abstract_and_Geometric_Algebra/First-Semester_Abstract_Algebra%3A_A_Structural_Approach_(Sklar)/02%3A_Groups/2.01%3A_Binary_Operations_and_StructuresSo far we have been discussing sets. These are extremely simple objects, essentially mathematical “bags of stuff.” Without any added structure, their usefulness is very limited. A set with no added st...So far we have been discussing sets. These are extremely simple objects, essentially mathematical “bags of stuff.” Without any added structure, their usefulness is very limited. A set with no added structure will not help us, say, solve a linear equation. What will help us with such things are objects such as groups, rings, fields, and vector spaces. These are sets equipped with binary operations which allow us to combine set elements in various ways.
- https://math.libretexts.org/Bookshelves/Abstract_and_Geometric_Algebra/First-Semester_Abstract_Algebra%3A_A_Structural_Approach_(Sklar)/02%3A_Groups/2.04%3A_Examples_of_Groups%2F%2FNongroups%2C_Part_ILet's look at some examples of groups/nongroups.
