IsomorphismEdexcel A-Level Further Maths: Revision notes

Two groups $(G,*)$ and $(H,\circ)$ are isomorphic, written $G\cong H$, if there is a function $\varphi:G\to H$ that is

1. a bijection (one-to-one and onto), and
2. operation-preserving: $\varphi(a*b)=\varphi(a)\circ\varphi(b)$ for all $a,b\in G$. Isomorphic groups have the same structure: they differ only in the names of the elements and the symbol for the operation. An isomorphism sends identity to identity, $\varphi(e_G)=e_H$, and inverses to inverses, $\varphi(a^{-1})=\varphi(a)^{-1}$. At this level isomorphisms are restricted to groups of order at most 8.

Give an explicit mapping $\varphi$, then show (i) it is a bijection (list the images and check that they are all distinct and cover $H$) and (ii) it preserves the operation, for general $a,b$ or by checking every pair against the Cayley tables. Example: $G=\{1,i,-1,-i\}$ under multiplication and $H=\{0,1,2,3\}$ under addition mod 4. Define $\varphi(i^k)=k$, so $1\mapsto0$, $i\mapsto1$, $-1\mapsto2$, $-i\mapsto3$. Then $\varphi(i^a\cdot i^b)=\varphi(i^{a+b})=a+b\bmod4=\varphi(i^a)+\varphi(i^b)$. A cyclic group of order $n$ is isomorphic to $Z_n$: map the generator $g$ to $1$, so $\varphi(g^k)=k$.

If $G\cong H$ then they have:

• the same order;
• the same number of elements of each order, because $\varphi(g^n)=\varphi(g)^n$;
• both cyclic or both non-cyclic;
• both abelian or both non-abelian. The order of $\varphi(g)$ equals the order of $g$, so an element can only be mapped to an element of the same order. This restricts the possible mappings: in $G=\{1,i,-1,-i\}$ the element $i$ has order $4$, so it can only map to $1$ or $3$ in $Z_4$.

To show $G\not\cong H$, find one property that differs. The usual choices are:

• different orders (different numbers of elements);
• different numbers of elements of some order, e.g. $K=\{1,3,5,7\}$ mod 8 has three elements of order $2$ but $Z_4$ has one;
• one group cyclic, the other not (no element of order $|G|$);
• one group abelian, the other not, e.g. $Z_6$ and $S_3$. You do not need to show that every possible bijection fails: a single differing property is enough.

For small orders only a few structures exist:

• Order $2$, $3$, $5$, $7$ (primes): one structure, the cyclic group.
• Order $4$: the cyclic group $Z_4$ (for example $\{1,i,-1,-i\}$ or the rotations of a square) and the group where every element has order at most 2, such as $\{1,3,5,7\}$ mod 8.
• Order $6$: the cyclic group $Z_6$ and the non-abelian group $S_3$ (symmetries of an equilateral triangle). Use the properties above to decide which type a given group of order up to 8 belongs to.