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 inverses in it.
- Find the order of a group and of an element, and recognise cyclic groups.
WHAT YOU PROBABLY THINK
Group theory is about numbers, so every group is really arithmetic under another name.
Four rules
A group is a set G with a binary operation satisfying four axioms: closure, so the operation never leaves the set; associativity; an identity element leaving everything unchanged; and an inverse for every element. Nothing in that list mentions numbers, and the opening claim assumes what the axioms deliberately leave out: 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 each a has inverse 5 − a. Four axioms, four ticks.
The complex set: products stay inside (i × i = −1, and so on), multiplication is associative, 1 is the identity, and each element has an inverse in the set.
Both have four or five elements and no arithmetic in common, 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 column, a Latin square, which is a quick sanity check on any table you build. The order of a group is how many elements it has; the order of an element is the smallest positive power returning 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 order divides 6, which is no accident: it is Lagrange's theorem showing up early.
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 can divide a prime.
YOUR TURN
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: every product reduces to one of 1, 2, 3, 4, since 5 is prime and none of these is a multiple of it. 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: every element, so 2 is a generator and the group is cyclic.
THE EXAM BIT
- Check the four axioms in order and name each one; a list of ticks without names earns little.
- Closure is the axiom most often missed, and the one most often broken by a proposed set.
- 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.
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
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A group needs four axioms: closure, associativity, an identity, and an inverse for every element.
Order of a group is its size; order of an element is the least power reaching the identity, and it always divides the group's order.
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.
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 inverses in it.
- Find the order of a group and of an element, and recognise cyclic groups.
Open the full revision checklist to see every objective in the course in one place.
No animated video for this topic yet; these notes stand alone.