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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/2.04%3A_Examples_of_Groups%2F%2FNongroups%2C_Part_ILet's look at some examples of groups/nongroups.
- https://math.libretexts.org/Bookshelves/Abstract_and_Geometric_Algebra/First-Semester_Abstract_Algebra%3A_A_Structural_Approach_(Sklar)/06%3A_Permutation_and_Dihedral_Groups/6.03%3A_Alternating_GroupsIn this section, we will discuss alternating groups and corresponding theorems.
