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
serg [7]
3 years ago
13

(a) If G is a finite group of even order, show that there must be an element a = e, such that a−1 = a (b) Give an example to sho

w that the conclusion of part (a) above does not hold for groups of odd order.
Mathematics
1 answer:
Dahasolnce [82]3 years ago
7 0

Answer:

See proof below

Step-by-step explanation:

First, notice that if a≠e and a^-1=a, then a²=e (this is an equivalent way of formulating the problem).

a) Since G has even order, |G|=2n for some positive number n. Let e be the identity element of G. Then A=G\{e} is a set with 2n-1 elements.

Now reason inductively with A by "pairing elements with its inverses":

List A as A={a1,a2,a3,...,a_(2n-1)}. If a1²=e, then we have proved the theorem.

If not, then a1^(-1)≠a1, hence a1^(-1)=aj for some j>1 (it is impossible that a^(-1)=e, since e is the only element in G such that e^(-1)=e). Reorder the elements of A in such a way that a2=a^(-1), therefore a2^(-1)=a1.

Now consider the set A\{a1,a2}={a3,a4,...,a_(2n-1)}. If a3²=e, then we have proved the theorem.

If not, then a3^(-1)≠a1, hence we can reorder this set to get a3^(-1)=a4 (it is impossible that a^(-1)∈{e,a1,a2} because inverses are unique and e^(-1)=e, a1^(-1)=a2, a2^(-1)=a1 and a3∉{e,a1,a2}.

Again, consider A\{a1,a2,a3,a4}={a5,a6,...,a_(2n-1)} and repeat this reasoning. In the k-th step, either we proved the theorem, or obtained that a_(2k-1)^(-1)=a_(2k)

After n-1 steps, if the theorem has not been proven, we end up with the set A\{a1,a2,a3,a4,...,a_(2n-3), a_(2n-2)}={a_(2n-1)}. By process of elimination, we must have that a_(2n-1)^(-1)=a_(2n-1), since this last element was not chosen from any of the previous inverses. Additionally, a_(2n1)≠e by construction. Hence, in any case, the statement holds true.

b) Consider the group (Z3,+), the integers modulo 3 with addition modulo 3. (Z3={0,1,2}). Z3 has odd order, namely |Z3|=3.

Here, e=0. Note that 1²=1+1=2≠e, and 2²=2+2=4mod3=1≠e. Therefore the conclusion of part a) does not hold

You might be interested in
HELP!! Simplify the expression 3y^2 • 4x^2y • 5x Using the laws of exponents. Show each step of your work.
JulsSmile [24]
<span>3y^2 • 4x^2y • 5x = (3*4*5) * (</span><span>y^2 * y) * (</span><span>x^2*x) = 60 * y^3 * x^3 = 60 (xy)^3


</span>where ((<span>y^2 * y)) adding the powers in case of multiplication

and also for this:</span><span> (<span>x^2*x)</span>  </span>
5 0
3 years ago
Find the measure of ∠. Show your work.
poizon [28]

Answer:

77

Step-by-step explanation:

If you have any questions about the way I solved,don't hesitate to ask

5 0
3 years ago
If a product of three numbers is 3150, two of the numbers are 14,15. Find the third number
kogti [31]

Answer:

15

proof on pic......................

3 0
3 years ago
Lincoln buys milk and oranges at the store.
podryga [215]

Answer: $6.24

Step-by-step explanation:

41.63-4.19= 37.44

37.44/6=6.24

8 0
3 years ago
Read 2 more answers
Y = - |x+3| - 4<br><br><br> It needs to be graphed, please help!
MA_775_DIABLO [31]

Answer:

Step-by-step explanation:

7 0
3 years ago
Read 2 more answers
Other questions:
  • When a snake hatched 4 years ago, it was only 5 inches long. Suppose it is now 3 foot 9 inches long. Given that the annual perce
    14·2 answers
  • Calculate the exact value of <br>26.32+38.8/13.10- 11.56​
    10·2 answers
  • Find the x-intercept and the y-intercept of the graph of the equation 1/3x + 9y = 15. Then graph the equation.
    12·2 answers
  • What is the perimeter of the pool? Need answer quickly
    12·1 answer
  • What is the quadratic formula?
    12·1 answer
  • Zoom in to see more clear :)
    11·1 answer
  • Please I need help badly please no links or I will report you
    13·1 answer
  • HHHHHHEEEEEEEELLLLLLLLLLLLLLLLPPPPPPPPPPPPPP MMMMMMMMMMMMMMMEEEEEEEEEEEEEEEEE WILL GIVE BRAINLESSS
    15·2 answers
  • A jet travels 650 miles in 3 hours. how far could the jet fly in 9 hours
    13·1 answer
  • 75 POINTS !!!!! URNGETNM EASY MATH PLEAS EIM CYRING I NEED NOW PLEAS ESOW ALL WORK AND EXPLAIN PLEASE
    11·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!