TechTorch

Location:HOME > Technology > content

Technology

Proving Cyclic Groups under Isomorphism

May 30, 2025Technology1688
Proving Cyclic Groups under Isomorphism The concept of isomorphism in

Proving Cyclic Groups under Isomorphism

The concept of isomorphism in group theory is fundamental for understanding the structural similarities between different groups. Specifically, if (G) is a cyclic group and (f: G to G') is a group isomorphism, then (G') is also cyclic. In this article, we will explore the proof step-by-step and understand the significance of this property in the broader context of group theory.

Definition of Cyclic Groups

A group (G) is cyclic if there exists an element (g in G) such that every element of (G) can be expressed as a power of (g). In terms of mathematical notation, this is expressed as (G langle g rangle).

Existence of a Generator

If (G) is cyclic, then there exists an element (g in G) such that every element in (G) can be written as (g^n) for some integer (n).

Applying the Isomorphism

Given that (f: G to G') is a group isomorphism, it preserves the group operation. This means that for any integer (n), the following must hold:

[f(g^n) (fg)^n]

This property is crucial because it allows us to relate the generators and elements of both groups through the isomorphism (f).

Generating (G')

Define (h fg). We need to show that (h) generates (G'). Take any element (y in G'). Since (f) is an isomorphism, there exists some (x in G) such that:

[f(x) y]

Because (G) is cyclic and (g) is a generator, we can express (x) as (x g^k) for some integer (k). Thus, we have:

[f(g^k) f(g)^k h^k]

This implies:

[f(x) f(g^k) h^k y]

Hence, every element (y in G') can be represented as (h^k) for some integer (k). Therefore, (h) generates (G') and (G') is cyclic.

Alternative Proof

Alternatively, if (a) is a generator of (G), then (fa) is a generator of (G'). Given any (x in G), it has a preimage (y in G) such that (fy x). Since (a) is a generator of the cyclic group (G), we can express (y) as (y a^k) for some integer (k). Thus:

[fy fa^k fa^k]

Therefore, every (x in G) is an integer power of (fa), implying that (G') is cyclic and generated by (fa).

Conclusion

We have shown that if (G) is a cyclic group and (f: G to G') is a group isomorphism, then (G') is also cyclic. This result highlights the importance of isomorphisms in preserving the structural properties of groups, particularly in the case of cyclic groups.