Listing methods to prove that two groups are not isomorphic

Basically any property that is preserved by group isomorphisms will do. This includes:

  • order of elements
  • commutativity
  • amount of subgroups of a certain (finite) order
  • amount of Sylow p-subgroups
  • cardinality of the group
  • being cyclic (2 cyclic groups of same cardinality are isomorphic)
  • order of subgroups

Another method (for finite groups) is to look at character tables. If two groups have different character tables, they can't be isomorphic. Notice that the converse is false: two groups can have the same character table without being isomorphic (If I remember correctly $D_8$ and $Q_8$ are a counterexample when considering the character table for the complex irreducibel representations, but correct me if I'm wrong!)

In practice, one can look at special subgroups like the center and normalisator of a group, as these are often easier to understand than the entire group.


  1. The automorphism group $Aut(G)$

  2. The cohomology groups $H^n(G,M)$ for $G$-modules $M$

  3. The homology groups $H_n(G,M)$ for $G$-modules $M$

  4. Solvability, Supersolvability, Nilpotency


Here are two more:

  • Number of generators. To be more precise: you can prove that two groups are not isomorphic by proving that one of them is spanned by a set with a certain cardinal, whereas no set with that cardinal spans the other one.
  • Quotient groups: if $G$ and $H$ are isomorphic and $N$ is a normal subgroup of $G$, then there has to be a normal subgroup $M$ of $H$ such that $G/N\simeq H/M$.

Tags:

Group Theory

Big List

Group Isomorphism

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