Questions about Z × Other invariants of Z × Distribution of ω ( n ) Elements and subgroups n n Anatomy of the mulitplicative group Greg Martin University of British Columbia Pacific Northwest Number Theory Conference University of Idaho May 19, 2012 slides can be found on my web page www.math.ubc.ca/ ∼ gerg/index.shtml?slides Anatomy of the mulitplicative group Greg Martin
Questions about Z × Other invariants of Z × Distribution of ω ( n ) Elements and subgroups n n Outline What do we want to know about multiplicative 1 groups (mod n ) ? Distribution of the number of prime factors of n 2 Other invariants of the multiplicative groups 3 Counting certain elements, and subgroups, of multiplicative 4 groups Anatomy of the mulitplicative group Greg Martin
Questions about Z × Other invariants of Z × Distribution of ω ( n ) Elements and subgroups n n The main characters Notation The quotient ring Z / n Z will be denoted by Z n . It enjoys both addition and multiplication. If we ignore multiplication: The additive group Z + n is the set Z n with the ring’s addition operation. It is always a cyclic group with n elements. If we instead ignore addition: n is the set ( Z n ) × of invertible The multiplicative group Z × elements in Z n , with the ring’s multiplication operation. It is a finite abelian group with φ ( n ) elements. Anatomy of the mulitplicative group Greg Martin
Questions about Z × Other invariants of Z × Distribution of ω ( n ) Elements and subgroups n n The main characters One ring to rule them all . . . The quotient ring Z / n Z will be denoted by Z n . It enjoys both addition and multiplication. If we ignore multiplication: The additive group Z + n is the set Z n with the ring’s addition operation. It is always a cyclic group with n elements. If we instead ignore addition: n is the set ( Z n ) × of invertible The multiplicative group Z × elements in Z n , with the ring’s multiplication operation. It is a finite abelian group with φ ( n ) elements. Anatomy of the mulitplicative group Greg Martin
Questions about Z × Other invariants of Z × Distribution of ω ( n ) Elements and subgroups n n The main characters One ring to rule them all . . . The quotient ring Z / n Z will be denoted by Z n . It enjoys both addition and multiplication. If we ignore multiplication: The additive group Z + n is the set Z n with the ring’s addition operation. It is always a cyclic group with n elements. If we instead ignore addition: n is the set ( Z n ) × of invertible The multiplicative group Z × elements in Z n , with the ring’s multiplication operation. It is a finite abelian group with φ ( n ) elements. Anatomy of the mulitplicative group Greg Martin
Questions about Z × Other invariants of Z × Distribution of ω ( n ) Elements and subgroups n n The main characters One ring to rule them all . . . The quotient ring Z / n Z will be denoted by Z n . It enjoys both addition and multiplication. If we ignore multiplication: The additive group Z + n is the set Z n with the ring’s addition operation. It is always a cyclic group with n elements. If we instead ignore addition: n is the set ( Z n ) × of invertible The multiplicative group Z × elements in Z n , with the ring’s multiplication operation. It is a finite abelian group with φ ( n ) elements. Anatomy of the mulitplicative group Greg Martin
Questions about Z × Other invariants of Z × Distribution of ω ( n ) Elements and subgroups n n How to find the structure of Z × n Chinese remainder theorem, and primitive roots If the prime factorization of n is p r 1 1 × · · · × p r k k , then Z × n ∼ = Z × 1 × · · · × Z × ∼ = Z + ( p 1 − 1 ) × · · · × Z + ( p k − 1 ) . p r 1 rk p r 1 − 1 rk − 1 p p k 1 k Confession I’m lying slightly about the prime p = 2 . I’ll keep doing so throughout the talk when making general statements about Z × n . Example (with n = 11 ! ) 11 ! ∼ Z × = Z × 2 8 ⊕ Z × 3 4 ⊕ Z × 5 2 ⊕ Z × 7 ⊕ Z × 11 ∼ = ( Z + 2 ⊕ Z + 64 ) ⊕ Z + 54 ⊕ Z + 20 ⊕ Z + 6 ⊕ Z + 10 Anatomy of the mulitplicative group Greg Martin
Questions about Z × Other invariants of Z × Distribution of ω ( n ) Elements and subgroups n n How to find the structure of Z × n Chinese remainder theorem, and primitive roots If the prime factorization of n is p r 1 1 × · · · × p r k k , then Z × n ∼ = Z × 1 × · · · × Z × ∼ = Z + ( p 1 − 1 ) × · · · × Z + ( p k − 1 ) . p r 1 rk p r 1 − 1 rk − 1 p p k 1 k Confession I’m lying slightly about the prime p = 2 . I’ll keep doing so throughout the talk when making general statements about Z × n . Example (with n = 11 ! ) 11 ! ∼ Z × = Z × 2 8 ⊕ Z × 3 4 ⊕ Z × 5 2 ⊕ Z × 7 ⊕ Z × 11 ∼ = ( Z + 2 ⊕ Z + 64 ) ⊕ Z + 54 ⊕ Z + 20 ⊕ Z + 6 ⊕ Z + 10 Anatomy of the mulitplicative group Greg Martin
Questions about Z × Other invariants of Z × Distribution of ω ( n ) Elements and subgroups n n How to find the structure of Z × n Chinese remainder theorem, and primitive roots If the prime factorization of n is p r 1 1 × · · · × p r k k , then Z × n ∼ = Z × 1 × · · · × Z × ∼ = Z + ( p 1 − 1 ) × · · · × Z + ( p k − 1 ) . p r 1 rk p r 1 − 1 rk − 1 p p k 1 k Confession I’m lying slightly about the prime p = 2 . I’ll keep doing so throughout the talk when making general statements about Z × n . Example (with n = 11 ! ) 11 ! ∼ Z × = Z × 2 8 ⊕ Z × 3 4 ⊕ Z × 5 2 ⊕ Z × 7 ⊕ Z × 11 ∼ = ( Z + 2 ⊕ Z + 64 ) ⊕ Z + 54 ⊕ Z + 20 ⊕ Z + 6 ⊕ Z + 10 Anatomy of the mulitplicative group Greg Martin
Questions about Z × Other invariants of Z × Distribution of ω ( n ) Elements and subgroups n n How to find the structure of Z × n Chinese remainder theorem, and primitive roots If the prime factorization of n is p r 1 1 × · · · × p r k k , then Z × n ∼ = Z × 1 × · · · × Z × ∼ = Z + ( p 1 − 1 ) × · · · × Z + ( p k − 1 ) . p r 1 rk p r 1 − 1 rk − 1 p p k 1 k Confession I’m lying slightly about the prime p = 2 . I’ll keep doing so throughout the talk when making general statements about Z × n . Example (with n = 11 ! ) 11 ! ∼ Z × = Z × 2 8 ⊕ Z × 3 4 ⊕ Z × 5 2 ⊕ Z × 7 ⊕ Z × 11 ∼ = ( Z + 2 ⊕ Z + 64 ) ⊕ Z + 54 ⊕ Z + 20 ⊕ Z + 6 ⊕ Z + 10 Anatomy of the mulitplicative group Greg Martin
Questions about Z × Other invariants of Z × Distribution of ω ( n ) Elements and subgroups n n How to find the structure of Z × n Chinese remainder theorem, and primitive roots If the prime factorization of n is p r 1 1 × · · · × p r k k , then Z × n ∼ = Z × 1 × · · · × Z × ∼ = Z + ( p 1 − 1 ) × · · · × Z + ( p k − 1 ) . p r 1 rk p r 1 − 1 rk − 1 p p k 1 k Confession I’m lying slightly about the prime p = 2 . I’ll keep doing so throughout the talk when making general statements about Z × n . Example (with n = 11 ! ) 11 ! ∼ Z × = Z × 2 8 ⊕ Z × 3 4 ⊕ Z × 5 2 ⊕ Z × 7 ⊕ Z × 11 ∼ = ( Z + 2 ⊕ Z + 64 ) ⊕ Z + 54 ⊕ Z + 20 ⊕ Z + 6 ⊕ Z + 10 Anatomy of the mulitplicative group Greg Martin
Questions about Z × Other invariants of Z × Distribution of ω ( n ) Elements and subgroups n n How to find the structure of Z × n Chinese remainder theorem, and primitive roots If the prime factorization of n is p r 1 1 × · · · × p r k k , then Z × n ∼ = Z × 1 × · · · × Z × ∼ = Z + ( p 1 − 1 ) × · · · × Z + ( p k − 1 ) . p r 1 rk p r 1 − 1 rk − 1 p p k 1 k Confession I’m lying slightly about the prime p = 2 . I’ll keep doing so throughout the talk when making general statements about Z × n . Example (with n = 11 ! ) 11 ! ∼ Z × = Z × 2 8 ⊕ Z × 3 4 ⊕ Z × 5 2 ⊕ Z × 7 ⊕ Z × 11 ∼ = ( Z + 2 ⊕ Z + 64 ) ⊕ Z + 54 ⊕ Z + 20 ⊕ Z + 6 ⊕ Z + 10 Anatomy of the mulitplicative group Greg Martin
Questions about Z × Other invariants of Z × Distribution of ω ( n ) Elements and subgroups n n Two standard forms Invariant factors Every finite abelian group G is uniquely isomorphic to a direct sum of cyclic groups G ∼ = Z + d 1 ⊕ Z + d 2 ⊕ · · · ⊕ Z + d m where d 1 | d 2 | · · · | d m . Example (with n = 11 ! ) Z × 11 ! ∼ = Z + 2 ⊕ Z + 2 ⊕ Z + 2 ⊕ Z + 2 ⊕ Z + 60 ⊕ Z + 8640 Primary decomposition Every finite abelian group G is isomorphic to a direct sum of cyclic groups G ∼ = Z + 1 ⊕ Z + 2 ⊕ · · · ⊕ Z + ℓ where each q j is q r 1 q r 2 r ℓ q prime. (unique up to reordering) Anatomy of the mulitplicative group Greg Martin
Questions about Z × Other invariants of Z × Distribution of ω ( n ) Elements and subgroups n n Two standard forms Invariant factors Every finite abelian group G is uniquely isomorphic to a direct sum of cyclic groups G ∼ = Z + d 1 ⊕ Z + d 2 ⊕ · · · ⊕ Z + d m where d 1 | d 2 | · · · | d m . Example (with n = 11 ! ) Z × 11 ! ∼ = Z + 2 ⊕ Z + 2 ⊕ Z + 2 ⊕ Z + 2 ⊕ Z + 60 ⊕ Z + 8640 Primary decomposition Every finite abelian group G is isomorphic to a direct sum of cyclic groups G ∼ = Z + 1 ⊕ Z + 2 ⊕ · · · ⊕ Z + ℓ where each q j is q r 1 q r 2 r ℓ q prime. (unique up to reordering) Anatomy of the mulitplicative group Greg Martin
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