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, of G, we have xy = yx Note that e is an element of G such that ex = x Also note that for all the above, the group operation is not necessarily multiplication.

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=ab vi. (R*,) where R* is the set of non-zero reals and where a◦b =ab 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+b2  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 aG 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 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. ... click for more

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 H of G 3. show that if a elements in G where G is a finite group with the identity, e, then there exist n elements in Z+ such that a ⁿ =e 4. 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. 5. if... click for more