Maths › Further Pure 2 › Groups and their axioms
Groups and their axioms
Four rules about a set and an operation, and a whole branch of algebra opens. Symmetries, clock arithmetic and matrices turn out to be the same structure written out in different alphabets.
Builds on Matrix algebra and transformations and The structure of proof.
IN THIS TOPIC
- State the four group axioms and test whether a set and operation satisfy them.
- Build and read a Cayley table, and spot the identity and the inverses in it.
- Find the order of a group and the order of an element.
- Recognise cyclic groups and name a generator.
COMMON MISCONCEPTION
Closure just means the answer lies somewhere sensible, in the universal set if nowhere else.
Four rules
A group is a set G with a binary operation satisfying four axioms. Closure keeps the operation inside the set. Associativity lets you bracket however you like. An identity element leaves everything unchanged, and every element needs an inverse. Closure is stricter than answers merely existing somewhere: the product of two elements of G must itself lie in G, and landing in some larger set fails the axiom. Nothing in the list mentions numbers, either. The elements can be rotations, matrices or permutations, and the theory does not care.
WORKED EXAMPLE
Three groups that look nothing alike
Verify that the integers modulo 5 under addition, and the set {1, −1, i, −i} under multiplication, are both groups.
Modulo 5: sums stay in the set, addition is associative, 0 is the identity, and a has inverse 5 − a reduced modulo 5. Four axioms, four ticks.
The complex set: products stay inside, since i × i = −1 and so on, multiplication is associative, 1 is the identity, and each element has an inverse in the set.
One set has five elements and the other four, and they share no arithmetic at all, yet both satisfy the same four rules. The rotational symmetries of a square make a third example with the same structure as the second.
Cayley tables and order
A Cayley table lists every product. The identity's row and column repeat the headings unchanged, and each element appears exactly once in every row and every column. That Latin square property is a quick sanity check on any table you build. The order of a group is how many elements it has, and the order of an element is the smallest positive power that returns the identity.
WORKED EXAMPLE
Orders inside a small group
In the integers modulo 6 under addition, find the order of each element.
0 has order 1. Adding 1 repeatedly needs six steps to reach 0, so 1 has order 6, and so does 5.
2 + 2 + 2 = 0, so 2 has order 3, as does 4. And 3 + 3 = 0, so 3 has order 2.
Every one of those orders divides 6. That is no accident; it is Lagrange's theorem showing up early.
Cyclic groups, and what the axioms do not require
A group generated by a single element is cyclic. The integers modulo n under addition always are, generated by 1. In a cyclic group of prime order every element except the identity is a generator, since no smaller order could divide a prime.
Notice what the axioms never asked for. There is no requirement that the operation commutes, and plenty of groups do not. The six symmetries of an equilateral triangle form a group in which a reflection followed by a rotation differs from the same pair the other way round. A group that does commute is called abelian, and it is worth saying so explicitly when a question turns on the distinction.
GUIDED PRACTICE
Is it a group?
Decide whether the set {1, 2, 3, 4} under multiplication modulo 5 forms a group, and if so, name a generator.
Show the working
Closure holds, since 5 is prime and none of these is a multiple of it, so every product reduces to one of 1, 2, 3, 4. Multiplication is associative and 1 is the identity.
Inverses: 2 × 3 = 6 = 1, so 2 and 3 are mutual inverses; 4 × 4 = 16 = 1, so 4 is its own; 1 is its own.
It is a group of order 4. Powers of 2 give 2, 4, 3, 1, which is every element, so 2 is a generator and the group is cyclic.
ASSESSMENT FOCUS
- Check the four axioms in order and name each one; a column of ticks without names earns little.
- Closure is the axiom most often skipped, and the one a proposed set most often fails.
- In a Cayley table, every row and column must be a rearrangement of the elements. Use it to catch errors.
- The order of an element is the smallest positive power giving the identity, not any power that works.
- Say whether a group is abelian when the question mentions commuting, and give a counterexample if it is not.
CHECK YOURSELF
In the group of integers modulo 8 under addition, find the order of the element 6.
Show a hint
Add 6 repeatedly modulo 8 until you reach 0.
Show the answer
6, then 12 = 4, then 18 = 2, then 24 = 0. Four steps, so 6 has order 4. As Lagrange requires, 4 divides the group's order 8.
A group needs four axioms: closure, associativity, an identity, and an inverse for every element.
The order of a group is its size; the order of an element is the least power reaching the identity.
A group generated by one element is cyclic, and the integers modulo n under addition always are.
WORKBOOK
Printable practice for this topic: original exam-style questions with room to work, and a fully worked answer book. Free to use; please do not redistribute or sell.
Or read them with their worked answers on the groups and their axioms questions page.
CHECK YOUR PROGRESS
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- State the four group axioms and test whether a set and operation satisfy them.
- Build and read a Cayley table, and spot the identity and the inverses in it.
- Find the order of a group and the order of an element.
- Recognise cyclic groups and name a generator.
Open the full revision checklist to see every objective in the course in one place.