Higher Algebra I — Fall 2005

Gerald Hoehn

Problem sheet 1 August 25, 2005

Problem 1: Prove Lemma 2 from Chapter 1: In a commutative semigroup H one has ∏_{i=1}^{n} a_{i} = ∏_{i=1}^{n} a_{π(i)} for any a_{1}, …, a_{n} ∈ H and bijective map π : {1, …, n} ⟶ {1, …, n}.

Problem 2: A left identity in a semigroup G is an element e such that ea = a for all a ∈ G. A left inverse of a ∈ G is an element x ∈ G such that xa = e, with e a left identity. Prove that a semigroup which has a left identity and every element has a left inverse is in fact a group and the left identity is an identity and a left inverse is an inverse.

Problem 3: (i) Is the additive group of integers isomorphic to the additive group of rationals?

(ii) Is the additive group of rationals isomorphic to the multiplicative group of non-zero rationals?

Problem 4: Prove part (iv) and (v) of the list of subgroup examples:

(iv) H is a subgroup of the additive group of integers if and only if there exists a n ∈ Z such that H = nZ := {nk | k ∈ Z}.

(v) The set Q = \{±E, ±I, ±J, ±K\} with E = \binom{1 0}{0 1}, I = \binom{i 0}{0 −i}, J =

\binom{0 1}{−1 0}, K = \binom{0 i}{i 0}, is a subgroup of GL_{2}(C) of size 8.

Furthermore, determine all subgroups of the quaternion group Q.

Problem 5^{∗}: Generalize Theorem 3 of Chapter 1 to arbitrary (i.e., not necessary abelian) semigroups and groups.