Groups and subgroups

Concepts

• Subgroup 10 exercisesA subset H of a group G that is itself a group under the operation of G.
• Order of a group 10 exercisesThe order of a finite group G is its number of elements; if G is infinite it has infinite order.
• Order of an element 10 exercisesThe order of an element a of a group with identity e is the least integer n>0 with a^n=e; if there is none, a is said to have inf…
• Generated subgroup 10 exercisesThe subgroup generated by a subset S of a group G is the smallest subgroup H of G containing S; it is denoted langle Srangle and…
• Cyclic group 10 exercisesA cyclic group is a group G with an element g, called a generator, such that every element of G is a power of g.
• Center of a group 10 exercisesThe center of a group G is the set Z(G) of elements of G that commute with every element of G.
• Group homomorphism 10 exercisesA group homomorphism is a function φ from a group G to a group H that respects the operation.
• Group isomorphism 10 exercisesA group isomorphism is a bijective homomorphism φ from a group G to a group H; then G and H are isomorphic.