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Mathematics · Advanced university

Groups, subgroups and element order

An associative operation has an identity and an inverse for every element; commutativity is optional.

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Prerequisites

Proofs, sets and modular arithmetic.

Study notes

Group

An associative operation has an identity and an inverse for every element; commutativity is optional.

Subgroup

A nonempty subset closed under products and inverses inherits the group structure.

Element order

The least positive power returning the identity is the finite order; some elements have infinite order.

Lagrange

The order of a subgroup of a finite group divides the group order.

Worked examples

What is the order of [4] in the additive group Z/10Z?

The least positive k with 10 dividing 4k is 5; equivalently 10/gcd(4,10)=5.

5

Prove the identity element of a group is unique.

Credit both identity properties.

If e and f are both identities, then ef=f because e is an identity and ef=e because f is an identity, so e=f.

Check your understanding

  1. Must every group be commutative?

    1. Only infinite groups are groups
    2. No
    3. Yes by associativity
    4. Yes if it has an identity

  2. A finite group has 18 elements. A subgroup has 6 elements. How many left cosets does it have?

  3. Show that the set of even integers is a subgroup of the additive integers.

  4. Can a group of order 15 have a subgroup of order 4?

    1. Yes by Lagrange's theorem
    2. Always
    3. Only when abelian
    4. No

Answers and marking guidance

Answers and marking notes
  1. No

    Commutativity is an additional property defining an abelian group.

  2. 3

    The left cosets partition the group into equally sized sets of 6.

  3. It contains 0, sums of even integers are even, and additive inverses of even integers are even.

    Credit identity and closure under addition and inverses.

  4. No

    4 does not divide 15, so Lagrange rules it out.

Further reading and review references

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