Groups and subgroups · Definition

Group isomorphism

modern algebragroups

Before you start you need to know

An isomorphism says two groups are the same in different clothes: only the names of the elements change.

Definition

A group isomorphism is a bijective homomorphism \(\varphi\) from a group \(G\) to a group \(H\); then \(G\) and \(H\) are isomorphic.

\[G\cong H\]

If \(\varphi\colon G\to H\) is an isomorphism, its inverse \(\varphi^{-1}\) is one too, and the composition of isomorphisms is an isomorphism; hence \(\cong\) is an equivalence relation. Classic examples: \(x\mapsto e^x\) identifies \((\mathbb{R},+)\) with \((\mathbb{R}_{>0},\cdot)\), and \(\mathbb{Z}_6\cong\mathbb{Z}_2\times\mathbb{Z}_3\) because \(\gcd(2,3)=1\).

An isomorphism preserves everything defined through the operation: the order of the group, the order of each element, being abelian, being cyclic, the number of subgroups of each size. These invariants let us prove two groups are not isomorphic: it is enough that one has a property the other lacks.

Thus \(\mathbb{Z}_4\not\cong\mathbb{Z}_2\times\mathbb{Z}_2\) because only the first has an element of order \(4\), and \(\mathbb{Z}_6\not\cong S_3\) because only the first is abelian. Proving that two groups are isomorphic, instead, requires building the function \(\varphi\) and checking it is bijective and respects the operation.

An isomorphism from a group onto itself is called an automorphism. For \(\mathbb{Z}_n\) the automorphisms send \(1\) to a generator, so there are \(\varphi(n)\) (Euler's function). Classifying groups \"up to isomorphism\" is the central goal of the theory: there are two of order \(4\), one of order \(5\) and two of order \(6\).

Think about it

Are \((\mathbb{Z},+)\) and \((\mathbb{Q},+)\) isomorphic?

No: \(\mathbb{Z}\) is cyclic and \(\mathbb{Q}\) is not, since the integer multiples of a rational \(q\ne0\) never include \(q/2\), so no \(q\) generates all of \(\mathbb{Q}\).

Exercises 10

  1. Which of these groups is isomorphic to \((\mathbb{Z}_6,+)\)?
  2. If \(G\) and \(H\) are isomorphic, which of these statements can be false?
  3. What is the largest order of an element of \(\mathbb{Z}_2\times\mathbb{Z}_4\)?
  4. Are \(\mathbb{Z}_4\) and \(\mathbb{Z}_2\times\mathbb{Z}_2\) isomorphic?
  5. The function \(x\mapsto e^x\) is an isomorphism from \((\mathbb{R},+)\) onto \((\mathbb{R}_{>0},\cdot)\). What is its inverse isomorphism?
  6. How many isomorphisms \((\mathbb{Z}_6,+)\to(\mathbb{Z}_6,+)\) are there?
  7. Why are \(\mathbb{Z}_6\) and \(S_3\) not isomorphic, even though both have order 6?
  8. How many isomorphisms \((\mathbb{Z}_{10},+)\to(\mathbb{Z}_{10},+)\) are there?
  9. Which of these groups of order 12 is isomorphic to \(\mathbb{Z}_{12}\)?
  10. How many groups of order 4 exist up to isomorphism?

References

  • David S. Dummit, Richard M. Foote, Abstract Algebra, 3rd edition, Wiley, 2004. §1.6
  • Joseph A. Gallian, Contemporary Abstract Algebra, 9th edition, Cengage, 2017. cap. 6
  • John B. Fraleigh, A First Course in Abstract Algebra, 7th edition, Pearson, 2002. §3