Group axioms and examplesEdexcel A-Level Further Maths: Revision notes

A binary operation $*$ on a set $S$ combines any two elements of $S$ to give one element. The set $S$ is closed under $*$ if $a*b\in S$ for all $a,b\in S$. Example: $\{1,2,3,4,5\}$ under multiplication modulo 6 is not closed, because $2\times3\equiv0$ and $0$ is not in the set. Always check closure first: it is the axiom most often failing.

A set $G$ with a binary operation $*$ is a group if:

1. Closure: $a*b\in G$ for all $a,b\in G$.
2. Associativity: $(a*b)*c=a*(b*c)$ for all $a,b,c\in G$.
3. Identity: there is $e\in G$ with $a*e=e*a=a$ for all $a\in G$.
4. Inverses: for every $a\in G$ there is $a^{-1}\in G$ with $a*a^{-1}=a^{-1}*a=e$. The identity is unique, and each element has exactly one inverse. If the operation is also commutative ($a*b=b*a$ for all $a,b$), the group is called abelian; this is not required by the axioms.

• Integers modulo $n$ under addition, $\{0,1,\ldots,n-1\}$: identity $0$, the inverse of $a$ is $n-a$ (and $0$ is self-inverse). This is a group for every $n$.
• Integers modulo $n$ under multiplication: $\{1,\ldots,n-1\}$ is a group only when $n$ is prime, e.g. $\{1,\ldots,6\}$ mod 7; the elements must all be invertible, e.g. $\{1,3,5,7\}$ mod 8 is a group but $\{1,\ldots,5\}$ mod 6 is not (it is not closed).
• Non-singular matrices: $2\times2$ matrices with non-zero determinant under multiplication, with $I$ as identity and the matrix inverse as inverse. Singular matrices have no inverse, so they are excluded.
• Symmetries of a geometrical figure under composition, e.g. the six symmetries of an equilateral triangle (three rotations including the identity and three reflections).
• Permutation groups: the permutations of $\{1,2,3\}$ under composition, a group of order 6. State the order of composition, for example $xy$ means do $y$ first.

A Cayley table lists all products $a*b$ for a finite group, with the row element on the left. Use it to check axioms: closure holds if every entry is in the set; the identity is the element whose row and column repeat the headings; and each element appears exactly once in every row and every column (a Latin square), which gives inverses. If the table is symmetrical about the leading diagonal the group is abelian. Example, $\{1,3,5,7\}$ mod 8: the table has $1$ down the leading diagonal, since every element is its own inverse.

A group is cyclic if every element is a power (or multiple) of one element $g$, called a generator: $G=\{e,g,g^2,\ldots\}$. In additive notation, the elements are the multiples $g,2g,3g,\ldots$. Examples: $\{0,1,2,3,4,5\}$ under addition modulo 6 is generated by $1$ (and by $5$) but not by $2$, which produces only $0,2,4$. $\{1,\ldots,6\}$ under multiplication modulo 7 is generated by $3$: $3,2,6,4,5,1$. The group $\{1,3,5,7\}$ mod 8 is not cyclic, because every element squares to the identity.