By Mark Steinberger

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GROUPS: BASIC DEFINITIONS AND EXAMPLES 39 The matrix a is said to reﬂect the plane through the x-axis, and we say that a matrix is a reﬂection matrix if it is a product a · c with c ∈ SO(2). (It is possible to give a direct geometric deﬁnition of reﬂection using dot products, and then to prove that these geometric operations are induced by precisely the matrices just stated. ) The most important piece of information about rotations and reﬂections is the way they interrelate. 1. For θ ∈ R, we have Rθ · a = a · R−θ .

Use this to obtain an iterative procedure to calculate (m, n). 4. Implement the iterative procedure above by a computer program to calculate greatest common divisors. † 5. A subgroup G ⊂ R is called discrete if for each g ∈ G there is an open interval (a, b) ⊂ R such that (a, b) ∩ G = {g}. Show that every discrete subgroup of R is cyclic. 4 Finite Cyclic Groups: Modular Arithmetic Here, we develop Zn , the group of integers modulo n. 1. Let n > 0 be a positive integer. , k − l ∈ n ). 2. Congruence modulo n is an equivalence relation.

Show that every subgroup of Zn is cyclic. 11. Show that Zn has exactly one subgroup of order d for each d dividing n, and has no other subgroups. ) Deduce the following consequences. (a) Show that a group containing a subgroup isomorphic to Zk × Zk for k > 1 cannot be cyclic. (b) Show that if a prime p divides n, then Zn has exactly p − 1 elements of order p. (c) Let H and K be subgroups of Zn . Show that H ⊂ K if and only if |H| divides |K|. (d) Show that if d divides n, then Zn has exactly d elements of exponent d.