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Mathematics · lesson 1 of 3

Groups, subgroups and element order

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

Advanced university8 practice questionsNotes + worked examples
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Learn the ideas

Prerequisites: Proofs, sets and modular arithmetic.

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

Example 1

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

Example 2

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

8 questions

Decimals or fractions are accepted for numeric answers. Proofs and written responses use sample answers and require human marking. This is practice, not a protected examination.

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

Show answer and explanation

5

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

2 Prove the identity element of a group is unique.

Sample answer and marking guidance

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.

Credit both identity properties.

3 Must every group be commutative?

Show answer and explanation

No

Commutativity is an additional property defining an abelian group.

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

Show answer and explanation

3

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

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

Sample answer and marking guidance

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.

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

Show answer and explanation

No

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

7 Why does Lagrange's theorem not say that every divisor is the order of a subgroup in every finite group?

Sample answer and marking guidance

The theorem provides a necessary divisibility condition, not a general converse. Extra structure or other results are needed for existence.

Credit the distinction between necessary and sufficient conditions.

8 What is the order of the identity element?

Show answer and explanation

1

The first positive power of the identity is already the identity.

Further reading and review references

These are reference links, not copied exercises, institutional endorsement or proof of independent review.