1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
Brums [2.3K]
3 years ago
6

Prove that u(n) is a group under the operation of multiplication modulo n.

Mathematics
2 answers:
amm18123 years ago
6 0

Answer:

The answer is the proof so it is long.

The question doesn't define u(n), but it's not hard to guess.


Group G with operation ∘

For all a and b and c in G:

1) identity: e ∈ G, e∘a = a∘e = a,

2) inverse: a' ∈ G, a∘a' = a'∘a = e,

3) closed: a∘b ∈ G,

4) associative: (a∘b)∘c = a∘(b∘c),

5) (optional) commutative: a∘b = b∘a.


Define group u(n) for n prime is the set of integers 0 < i < n with operation multiplication modulo n.


If n isn't prime, we exclude from the group all integers which share factors with n.


Identity: e = 1. Clearly 1∘a = a∘1 = a. (a is already < n).


Closed: u(n) is closed for n prime. We must show that for all a, b ∈ u(n), the integer product ab is not divisible by n, so that ab ≢ 0 (mod n). Since n is prime, ab ≠ n. Since a < n, b < n, no factors of ab can equal prime n. (If n isn't prime, we already excluded from u(n) all integers sharing factors with n).


Inverse: for all a ∈ u(n), there is a' ∈ u(n) with a∘a' = 1. To find a', we apply Euclid's algorithm and write 1 as a linear combination of n and a. The coefficient of a is a' < n.


Associative and Commutative:

(a∘b)∘c = a∘(b∘c) because (ab)c = a(bc)

a∘b = b∘a because ab = ba.


katrin2010 [14]3 years ago
5 0

Answer:

The answer is the proof so it is long.

The question doesn't define u(n), but it's not hard to guess.


Group G with operation ∘

For all a and b and c in G:

1) identity: e ∈ G, e∘a = a∘e = a,

2) inverse: a' ∈ G, a∘a' = a'∘a = e,

3) closed: a∘b ∈ G,

4) associative: (a∘b)∘c = a∘(b∘c),

5) (optional) commutative: a∘b = b∘a.


Define group u(n) for n prime is the set of integers 0 < i < n with operation multiplication modulo n.


If n isn't prime, we exclude from the group all integers which share factors with n.


Identity: e = 1. Clearly 1∘a = a∘1 = a. (a is already < n).


Closed: u(n) is closed for n prime. We must show that for all a, b ∈ u(n), the integer product ab is not divisible by n, so that ab ≢ 0 (mod n). Since n is prime, ab ≠ n. Since a < n, b < n, no factors of ab can equal prime n. (If n isn't prime, we already excluded from u(n) all integers sharing factors with n).


Inverse: for all a ∈ u(n), there is a' ∈ u(n) with a∘a' = 1. To find a', we apply Euclid's algorithm and write 1 as a linear combination of n and a. The coefficient of a is a' < n.


Associative and Commutative:

(a∘b)∘c = a∘(b∘c) because (ab)c = a(bc)

a∘b = b∘a because ab = ba.


You might be interested in
Please help fast!!!!
skelet666 [1.2K]

Answer:

if arc CB is 135° Then m(<D) = 67.5

Arc m(<D)=5×67.5=337.5

337.5×2=675

4 0
3 years ago
Which phrase should be inserted on the line to corrextly compare two fractions use the models to halp​
ivolga24 [154]

Answer:

need the answer choices

Step-by-step explanation:

7 0
3 years ago
Juan makes 75% of the goals he attempted during soccer practice. If he makes 30 goals at practice, how many does he attempt?
m_a_m_a [10]

Answer:

22.5

Step-by-step explanation:

30/4=7.5

7.5 times 3 = 22.5

Mark brainliest plz!

5 0
2 years ago
A bex contains 5 plain pencis and 5 pens. A second box contains 5 color pencils and 3 crayons. One item from each box is chosen
timama [110]

Answer:

x=16 pens

Step-by-step explanation:

7 0
3 years ago
Plz help -&gt;
Paraphin [41]

Answer: ok so for 5 and 125 it is 65 and 25

for 1 and 9 it is 5 and 3 and for 4 and 9 it is 13/2 and 6

6 0
3 years ago
Other questions:
  • Which statement is true regarding the graph?
    9·1 answer
  • Answer+explanation plz and thank you! :) these are quadratic equations.
    6·1 answer
  • Ron and Pam each have 20 pennies Ron will put his pennies into 4 groups pam will put her pennies into 5 groups who will have mor
    5·2 answers
  • If mc020-1.jpg and mc020-2.jpg, what is the domain of mc020-3.jpg?
    12·2 answers
  • Determine which pizza is the better buy in each situation. The 10-inch diameter pizza for $8.99 or the 6-inch diameter pizza for
    8·2 answers
  • What is the kinetic energy, in joules, of an object that has a mass of 100 kilograms and moves with a velocity of 10 m/s?
    9·1 answer
  • Bryan and his son, and Elton and his son all went fishing. Bryan caught as many fish as his son and Elton caught three times as
    8·1 answer
  • 5(4x – 2x) + 12 = 4x – 2
    11·1 answer
  • Simplify -2(3x-10)=15​
    15·2 answers
  • 75 POINTS PLEASE RESPOND ASAP
    10·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!