You want to specify all possible group homomorphisms from the free group F({a, b}) — the free group on two generators — to the symmetric group S₃ (which has 6 elements). How many such homomorphisms exist?
AExactly one — the free group has minimal structure, so there is essentially one natural map to any group
B36 — by the universal property, a homomorphism from F({a,b}) to S₃ is uniquely determined by choosing where each generator maps, giving |S₃| × |S₃| = 6 × 6 = 36 independent choices
CNone — a free group cannot map to a finite group because it is infinite
D6 — one for each element of S₃, since the two generators must map to inverses of each other
The universal property of the free group, captured by the adjunction Hom_Grp(F(X), G) ≅ Hom_Set(X, U(G)), says that group homomorphisms from F(X) to G correspond bijectively to functions from the generating set X to the underlying set of G. With X = {a, b} and G = S₃, this gives |S₃| × |S₃| = 36 homomorphisms — one for each independent choice of where a and b map. There are no compatibility constraints because the free group imposes no relations: any function from generators extends uniquely to a homomorphism. This is the precise meaning of 'free.'
Question 2 Multiple Choice
The forgetful functor U: Grp → Set is faithful but not full. What does 'not full' mean in this context?
ASome group homomorphisms are lost — not every group homomorphism is remembered as a set function
BNot every function between the underlying sets of two groups is a group homomorphism — there are strictly more set morphisms between U(G) and U(H) than group morphisms between G and H
CThe forgetful functor can only be applied to finite groups, not infinite ones
DBeing not full means the functor is not surjective on objects — not every set is the underlying set of a group
Fullness would require every function f: U(G) → U(H) to be a group homomorphism — which is false. Most functions between underlying sets do not preserve the group operation. For example, the constant function mapping every element of ℤ to 0 in ℤ is a set function but fails to be a group homomorphism (f(1+1) = 0 ≠ 0 + 0 = f(1) + f(1) in the additive group). Faithfulness (injective on hom-sets) does hold because distinct homomorphisms remain distinct as set functions.
Question 3 True / False
Every group generated by a set X is a quotient of the free group F(X) on X.
TTrue
FFalse
Answer: True
By the universal property, any function X → G (sending generators to group elements) extends uniquely to a group homomorphism φ: F(X) → G. If X generates G, then φ is surjective. By the first isomorphism theorem, G ≅ F(X) / ker(φ) — G is a quotient of F(X) by the normal subgroup of relations. This is precisely the notion of a group presentation: G = ⟨X | R⟩ means G is the free group on X modded out by the normal closure of relations R. Free groups are universal; all other groups generated by X are obtained by imposing additional relations.
Question 4 True / False
Free objects are structurally trivial or minimal — for example, the free group on a single generator has mainly one element.
TTrue
FFalse
Answer: False
Free is the opposite of trivial. The free group on one generator {a} is the infinite cyclic group ℤ: the elements are {…, a⁻², a⁻¹, e, a, a², …} with no relations other than group axioms. The free group on two generators is an infinite non-abelian group — one of the most complex finitely generated groups. 'Free' means no extra relations beyond those required by the algebraic structure; this makes it the largest object generated by the given set, not the smallest. Every other group generated by one element (cyclic groups ℤ/nℤ) is a quotient of ℤ.
Question 5 Short Answer
Why does the adjunction Hom_Grp(F(X), G) ≅ Hom_Set(X, U(G)) precisely capture the universal property of free groups? What does this bijection tell you about how to define group homomorphisms out of a free group?
Think about your answer, then reveal below.
Model answer: The bijection says that specifying a group homomorphism from F(X) to any group G requires exactly as much data as specifying a function from the generating set X to the underlying set of G. Since F(X) has no relations, every element is a word in the generators, and the homomorphism property forces the image of each word to be determined by where the generators map. Conversely, any assignment of group elements to generators extends uniquely — there are no compatibility conditions to check because F(X) imposes no relations that could be violated. The adjunction packages this as a natural bijection: the free group turns set-theoretic data (where generators go) into algebraic data (a structure-preserving map). The universality is that this works for any group G without restriction.
The same pattern appears throughout algebra: free vector spaces (basis vectors map anywhere → linear maps), free monoids (generators map anywhere → monoid homomorphisms), polynomial rings (the indeterminate maps anywhere → ring homomorphisms). In each case, the free object is the left adjoint to the corresponding forgetful functor.