In a group G, which expression correctly gives the inverse of the product ab?
Aa⁻¹b⁻¹ — inverses of the individual elements in the same order
Bb⁻¹a⁻¹ — inverses of the individual elements in reversed order
Cb·a — elements in reversed order, uninverted
D(ab)⁻¹ cannot be simplified further without knowing the specific group
The socks-and-shoes rule states (ab)⁻¹ = b⁻¹a⁻¹ — the order reverses when taking the inverse of a product. You can verify directly: (ab)(b⁻¹a⁻¹) = a(bb⁻¹)a⁻¹ = a·e·a⁻¹ = aa⁻¹ = e. Option A is the most common error — students apply inverses element-by-element without reversing order, forgetting that non-commutative groups require the reversal. In an abelian (commutative) group, a⁻¹b⁻¹ = b⁻¹a⁻¹ happens to give the same result, but the correct general formula always reverses.
Question 2 Multiple Choice
In a group G, suppose ab = ac for elements a, b, c ∈ G. What can you conclude, and why?
ANothing — cancellation requires the group to be commutative (abelian)
Ba = e — only the identity can appear on both sides this way
Cb = c — left cancellation holds because every group element has an inverse
Db and c must both equal a⁻¹
Left cancellation holds in every group: multiply both sides on the left by a⁻¹ to get a⁻¹(ab) = a⁻¹(ac), then (a⁻¹a)b = (a⁻¹a)c by associativity, so e·b = e·c, therefore b = c. Crucially, this proof uses both the existence of inverses (a⁻¹ exists) and associativity — it does not require commutativity. In a monoid (associative with identity but no guaranteed inverses), cancellation may fail. Option A is the most tempting distractor: students confuse commutativity with cancellability.
Question 3 True / False
A group can have two distinct identity elements, as long as each satisfies the identity axiom independently.
TTrue
FFalse
Answer: False
The uniqueness of the identity is a theorem, not an additional axiom. If e and e' are both identities, then e = e·e' (because e' is an identity) = e' (because e is an identity). The two supposed identities are forced to be the same element. This brief proof is important: it shows the group axioms are not redundant — they constrain the structure enough to rule out multiple identities without explicitly requiring uniqueness.
Question 4 True / False
The cancellation law (if ab = ac then b = c) holds in any algebraic structure with a binary operation and an identity element, even without highly probable inverses.
TTrue
FFalse
Answer: False
Cancellation requires the existence of inverses, not just an identity. The proof multiplies both sides by a⁻¹, which only works if a⁻¹ exists. In a monoid (associative binary operation with identity, but no guaranteed inverses), cancellation can fail. For example, in the monoid of integers under multiplication, 0·3 = 0·5 but 3 ≠ 5, yet 0 has no multiplicative inverse. Groups guarantee inverses for all elements, which is what makes cancellation universally valid.
Question 5 Short Answer
Why is it necessary to prove that the identity element of a group is unique, rather than simply assuming uniqueness from the axiom that states an identity exists?
Think about your answer, then reveal below.
Model answer: The group axiom only states that at least one identity element exists — it says nothing about whether there could be more than one. Without a proof, it would be logically possible that different identities exist satisfying the axiom independently. The proof forces any two identities to be equal using only associativity and the identity property itself: if e and e' are both identities, then e = e·e' = e'. The axioms are sufficient to derive uniqueness, so adding it as a separate assumption would be redundant — but ignoring the proof would leave an unverified gap.
This same reasoning applies to inverse uniqueness: the axiom guarantees at least one inverse per element, but you must prove no element has two distinct inverses. These uniqueness proofs are not mere formalities — they establish that the algebraic structure is well-defined, with a single canonical identity and a single canonical inverse for each element, which every subsequent theorem in group theory depends on.