# Are Z21 and C2*C6 Isomorphic?

• razorg425
In summary, the conversation discusses how to prove that the units of Z21 are isomorphic to C2*C6. The conversation suggests starting by proving that Z21 is isomorphic to Z3*Z7, and then proving that the group on the right is cyclic. The Chinese Remainder Theorem is mentioned as a helpful tool in finding the necessary number n.
razorg425
Hi guys just a quick question on how I would go about showing the units of Z21 is isomorphic toC2*C6(cyclic groups).I have done out the multiplicative table but they seem to be different to me. What else can I do?

Start by proving that $$\mathbb{Z}_{21} \simeq \mathbb{Z}_3 \times \mathbb{Z}_7$$. What are the groups of units of $$\mathbb{Z}_3$$ and $$\mathbb{Z}_7$$?

(Hint: it's important that GCD(3,7)=1.)

Sorry Rochfor,
I actually can't prove that, I know it should be true as gcd(3,7)=1.
Thanks.

Try proving that the group on the right is cyclic.

Jeez i can't even do that.
Im having a terrible day with this.

So we want to show that every element of $$\mathbb{Z}_3 \times \mathbb{Z}_7$$ is of the form $$n \cdot ( [1]_3, [1]_7 )$$. So for $$x, y \in \mathbb{Z}$$, we want $$( [x]_3, [y]_7 ) = n \cdot ( [1]_3, [1]_7 ) = ( [n]_3, [n]_7 )$$. So we need to find a number n so that $$x \equiv n \mod 3$$ and $$y \equiv n \mod 7$$. The http://mathworld.wolfram.com/ChineseRemainderTheorem.html" is your friend.

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## 1. What is an isomorphic group?

An isomorphic group is a mathematical concept in abstract algebra where two groups have the same structure and are essentially the same, despite having different elements. In other words, isomorphic groups are groups that are equivalent to each other.

## 2. What is the group Z21?

The group Z21, also known as the cyclic group of order 21, is a mathematical group consisting of integers modulo 21 under addition. This means that the elements of the group are the integers from 0 to 20, and the group operation is addition modulo 21.

## 3. What is the group C2*C6?

The group C2*C6, also known as the direct product of C2 and C6, is a mathematical group formed by taking the Cartesian product of the cyclic group of order 2 (C2) and the cyclic group of order 6 (C6). This group contains elements of the form (a, b) where a belongs to C2 and b belongs to C6, and the group operation is defined as (a1, b1) * (a2, b2) = (a1 * a2, b1 * b2).

## 4. Are Z21 and C2*C6 isomorphic?

Yes, Z21 and C2*C6 are isomorphic groups. This is because both groups have the same number of elements (21), and they also have the same structure (cyclic groups of order 21). Therefore, they are essentially the same group, just with different elements.

## 5. How can we prove that Z21 and C2*C6 are isomorphic?

We can prove that Z21 and C2*C6 are isomorphic by finding a bijective homomorphism between the two groups. In other words, we need to find a function that maps each element of Z21 to an element of C2*C6 in a way that preserves the group operation. Once we establish this function, we can show that it is both one-to-one and onto, proving that the two groups are isomorphic.

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