Mathematics Homework Solutions
group theory
• I have the following cayley tables... see attached
Full description of alternating group A(4)
Need a full description of alternating group A(4), discussion of its subgroups (normal, sylow, cyclic), and their interrelationships. Any other details you can think of would be appreciated aswell.
Thanks
group proofs
Let G be a group. x and y are elements of G. Prove that:
a. The inverse of xy is y^-1x^-1
b. The identity element, e, is unique
c. The inverse of any element x of G is unique
d. If xy = xz then y = z
e. If x^-1y^-1=y^-1x^-1 then xy = yx
f. If every element x of G satisfies x x = e, then for any two elements, x, y, ...continues
group theory
1. each of the following is a possible group. For those not passing the test, list the group axiom or axioms that fail to hold.
i. (R,◦) where a ◦ b = aà—b
ii. (Z,◦) where a ◦ b = ab
iii. (Z,◦) where 2Z = 2n n Z and where a b=a+b
iv. ( R+,◦) where a◦b =ab
v. (Z,◦) where a◦b=ab
vi. (R*,) where R* is the set of non-zero reals and where a◦b =ab
2. let G be an abelian group and let cⁿ = c◦ ….◦c for n factors, where c g and n Z . use mathematical induction to prove that (a ◦ b)ⁿ = (a)ⁿ (b)ⁿ
3. which of the following are cyclic groups? For each that is , list all of its generators.
I (Z , + )
ii (6Z, + )
iii.( 6ⁿ n Z+,à— )
iv.( a+b2 a,b Z ,+)
v.( R, à— )
4. if (a ◦ b) ² =a² ◦ b², for elements a and b of a group G, then a ◦ b=b ◦ a, prove
5. let H be a subgroup of a group G such that g ֿ¹ hg H for all h H. Show every left coset gH is the same as the right coset Hg
6. prove that if G is an abelian group, written multiplicatively, with identity element e, then all elements, x, of G satisfying the equation x²=e form a sub group H of G
7. show that if aG where G is a finite group with the identity, e, then there exist n Z+ such that a ⁿ =e
8. prove the generalisation of the first part of this question: consider the set H of all solutions, x, of the equation x ⁿ =e for fixed integer n ≥1 in an abelian group, G with identity , e.
9. if ◦ is a binary operation on a set, S an element, x S is an idempotent for ◦ if x ◦ x= x. prove that a group has exactly one idempotent element.
10. define the term 'normal subgroup'. Give an example of a group, G, and a normal subgroup, H, of G
11. prove that every group, G, with identity, e, such that x ◦ x=e for all x G is abelian .
12. draw the cayley tables for the and V. for each group, list the pairs of inverses
13. determine whether the following are hohmorphisms. Let:
i. φ : Z → R under addition be given by φ (n) =n.
ii let G be any group and let : G→ G be given by φ (g)=g ֿ¹
for g G
Modern Algebra
Group Theory
Cyclic groups
...continues
GROUP THEORY
1. let H be a subgroup of a group G such that g ֿ¹ hg elements in H for all h elements in H. Show every left coset gH is the same as the right coset Hg
2. prove that if G is an abelian group, written multiplicatively, with identity element e, then all elements, x, of G satisfying the equation x²=e form a sub group ...continues
Group Theory
1.
i. State the axioms for an equivalence relation
ii. The relation n mod 3 divides the non-negative integers
(i.e, n in Z such that n ≥ 0) into how many partitions?
Show that n = 0 mod 3 is an equivalence relation.
2. Prove that, for any matrices, A, B and C:
A+B=B+A
And:
A+(B+C)=(A+B)+C
( i.e., that the matrix addition is both commutative and associative)
For simplicity, prove these properties using 2x2 matrices.
3. Prove that addition modulo n, written + is:
i. Associative.
ii. Commutative.
There are two ways to prove these properties. Each way requires a definition or two:
i. For n ≥ 2, 0 ≤ a, b ≤ n+1,
a+ b= a+b if a+b< n
a+n-n if a+b≥ n
ii. Writing a for a mod n and (a+b) = a+ b, then:
(p+ q) ≡ (p +q )
Do the proof using both methods. Which is more "algebraic" (in the sense of "abstract" algebra)?
4. Prove that addition modulo n, written + is:
i. Associative
ii Commutative.
( extra definations required : a for a mod n and (pà—q) = pà— q, so
(pà— q) (p à—q )
5.
i. State the axioms defining a group
- If (Z, +) is a group, show that it is. Identify the identity element for +; in addition, for each n in Z , identify the inverse. Also show that + is associative.
- If (Z, à—) is a group, show that it is. Identify the identity element for +; in addition, for each n in Z, identify the inverse. Also show that + is associative.
- If (R, à—) is a group, show that it is. Identify the identity element for +; in addition, for each n in Z , identify the inverse. Also show that + is associative.
iii. In each case, deterimine whether the algebraic structure is a group. For each such group:
- show how it satifies the group axioms
- Draw the cayley table for the group and list the inverse elements
i. For S=(0,2), a+2b ≡ (a+b) mod 2 and aà—2b ≡ (a à—b)mod2
a. (S,+ ) ( possibly an additive group)
b. (S,∙ 2) (possibly a multiplicative group).
ii. For S = (0,1,2) where n=2,3 and + and à— are defined as in the last part.
a. (S,+n) ( possibly an additive group)
b. (S,∙ n) (possibly a multiplicative group).
Determine whether any of the groups is an abelian group. If any of them are abelian:
i. state the conditions under which a group is abelian
ii. show that the group is abelian
6. there are only two groups of order four (Z 4and v). How many groups are there of the order five? Draw cayley tables for each one of them( the element should be named a,b,c,d,e) is either (or both) of the groups of order four subgroup of any of those of order five? If so which one
7. for each of the following structures, state whethere it is a group. If it is, state whether it is abelian or not.
i.For any set, A, the set of one-to one and onto functions, f: A →A under composition ( written "◦").
ii.The set of all subsets of the three-element set (a,b,c) ( there are eight such subsets) under:
a. Union
b. Intersection
iii. The set G=(a+b√5| a,b in Q) under addition and multiplication
iv The set consisting of non-zero numbers under
a. addition
b. division
v. The set (1,5,7,11) under multiplication modulo 12. Draw cayley table
vi. The set (4,6) under multiplication modulo 12. draw cayley table
vii. The set of real numbers under à—, where aà—b = 2(a+b)
viii The set of real numbers under +, where a+b = a+b-10
ix. The set of rotational symmetries of a regular hexagon under composition
x The following sets of permutations under composition
i. (e,(12),(123),(1234))
ii. (e,(12), (34), (12), (34))
8.Let G be a group, (G, * ) in which there is an element, a , such that g * g=g . prove that g=e
9.Prove that for every element, a, of a group, G, the order of a and a^-1 are the same ( including the case of an infinite order)
10.Let x and y be elements of a group, G. Prove that the elements xy and yx have the same orders
11.Find the subgroups of
i. Z7
ii. Z8
iii Z9
12.
i. determine which of the folowign are subgroups of under +
a. (0)
b. (-1,0,1)
c. (n| n=10m for some integer m
d.(p| p is a prime number
e. (0,1,2,3,4) under addition modulo 5
ii. Determine which of the following are subgroups of under mulitiplication:
a. (1, -1)
b. (x |x=3, for some integer n
c. (x |x=p/2ⁿ for some integers, p,n)
d. (x| x=k 3 for some interger k
Group Theory
Group Theory
1.
i. State the axioms for an equivalence relation
...continues
Homomorphism
2.Let G be abelian of order n. If gcd(n;m) = 1, prove that f(g) = gm is
an automorphism of G. (Note: Automorphism is just an isomorphism
from G to itself.)
3. If f : Z7 ! Z5 is a homomorphism, prove that f(x) = 0 for all x 2 Z7.
4. Prove that in the group S10 every permutation of order 20 must be odd.
5. Suppose G is a grou ...continues
Modern Algebra - #1
Let G be the octive group in the following table and let H be the subgroup H = {e, B} of G. Find the district right cosets of H in G, and write out their elements.
Please see attached for full question.
Modern Algebra - #2
Prove that mapping... is a homomorphism.
Note: both groups are under addition.
Please see attached for full question.
