# Groups, subgroups and element order

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

Course: [Abstract algebra and discrete proofs](https://studybrix.com/en/courses/algebra-and-discrete-proofs/index.md)
Level: advanced
Prerequisites: Proofs, sets and modular arithmetic.

## 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.

## Practice questions

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

Answer: 5

Explanation: 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 / 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.

Explanation: Credit both identity properties.

### 3. Must every group be commutative?

A. Only infinite groups are groups
B. No
C. Yes by associativity
D. Yes if it has an identity

Answer: No

Explanation: 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?

Answer: 3

Explanation: 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 / marking guidance: It contains 0, sums of even integers are even, and additive inverses of even integers are even.

Explanation: Credit identity and closure under addition and inverses.

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

A. Yes by Lagrange's theorem
B. Always
C. Only when abelian
D. No

Answer: No

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

Explanation: Credit the distinction between necessary and sufficient conditions.

### 8. What is the order of the identity element?

Answer: 1

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

## Further reading

- https://ocw.mit.edu/courses/18-100a-real-analysis-fall-2020/
- https://ocw.mit.edu/courses/18-06-linear-algebra-spring-2010/

Original AI-assisted educational draft; not independently teacher-reviewed. References support further study, not copied exercises, affiliation or curriculum accreditation.