Table of Contents
How do you prove a cyclic group is infinite?
A group G is cyclic if G = 〈g〉 for some g ∈ G. g is a generator of 〈g〉. If a generator g has order n, G = 〈g〉 is cyclic of order n. If a generator g has infinite order, G = 〈g〉 is infinite cyclic.
How do you show cyclic groups?
4 Answers. A finite group is cyclic if, and only if, it has precisely one subgroup of each divisor of its order. So if you find two subgroups of the same order, then the group is not cyclic, and that can help sometimes.
How do you prove a Homomorphism is an isomorphism?
If φ(G) = H, then φ is onto, or surjective. A homomorphism that is both injective and surjective is an an isomorphism. An automorphism is an isomorphism from a group to itself. If we know where a homomorphism maps the generators of G, we can determine where it maps all elements of G.
Does every group have a cyclic subgroup?
It is given that every element of a group generates a cyclic subgroup.
How do you find the generator of a cyclic group?
Every cyclic group is isomorphic to either Z or Z/nZ if it is infinite or finite. If it is infinite, it’ll have generators ±1. If it is finite of order n, any element of the group with order relatively prime to n is a generator. The number of relatively prime numbers can be computed via the Euler Phi Function ϕ(n).
How do you find the order of an element in a group?
The Order of an element of a group is the same as that of its inverse a-1. If a is an element of order n and p is prime to n, then ap is also of order n. Order of any integral power of an element b cannot exceed the order of b. If the element a of a group G is order n, then ak=e if and only if n is a divisor of k.
Why are cyclic groups important?
Note however that G is abelian. So the first non-abelian group has order six (equal to D3). One reason that cyclic groups are so important, is that any group G contains lots of cyclic groups, the subgroups generated by the ele ments of G. On the other hand, cyclic groups are reasonably easy to understand.
What is homomorphism and isomorphism of groups?
Isomorphism. A group homomorphism that is bijective; i.e., injective and surjective. Its inverse is also a group homomorphism. In this case, the groups G and H are called isomorphic; they differ only in the notation of their elements and are identical for all practical purposes.
What is a cyclic group of order n?
A cyclic group G G is a group that can be generated by a single element a a, so that every element in G G has the form ai a i for some integer i i. We denote the cyclic group of order n n by Zn Z n, since the additive group of Zn Z n is a cyclic group of order n n. Theorem: All subgroups of a cyclic group are cyclic.
How do you prove a cyclic group is cyclic?
Theorem: All subgroups of a cyclic group are cyclic. If G = ⟨ g ⟩ is a cyclic group of order n then for each divisor d of n there exists exactly one subgroup of order d and it can be generated by a n / d. Proof: Given a divisor d, let e = n / d. Let g be a generator of G.
What is an example of a cyclic subgroup?
Examples : Any a ∈ Z n ∗ can be used to generate cyclic subgroup ⟨ a ⟩ = { a, a 2,…, a d = 1 } (for some d ). For example, ⟨ 2 ⟩ = { 2, 4, 1 } is a subgroup of Z 7 ∗ . Any group is always a subgroup of itself. {1} is always a subgroup of any group. These last two examples are the improper subgroups of a group.
Does every group of composite order have proper subgroups?
Theorem: Every group of composite order has proper subgroups. Proof: Let G G be a group of composite order, and let 1 ≠ a ∈ G 1 ≠ a ∈ G . Then if ⟨a⟩ ≠ G ⟨ a ⟩ ≠ G we are done, otherwise the subgroup ⟨ad⟩ ≠ G ⟨ a d ⟩ ≠ G for every divisor d d of |G| | G |.