' Let (G, ) denote the set of all 2 × 2 real matrices A with det{A} # 0and det {A} EQ (the rational numbers).(a) Prove that (G, ·) is a group with respect to multiplication. (Matrix multiplicationalwaysassociative, so you may assume that. But check closure and the existence of an identity elementand inverse elements very carefully.)(b) Is this group Abelian? Justify.Given a group (G, *) and a nonempty set S. Let GS denote the set of allmappings from the set S to the set G. Find an operation on GS that will yield a group. Show that yourchoice of operation is correct.

a) Let A, B ∈ ( G, ·) ⇒ det {A} ≠ 0, det {A} ∈ QAlso det {B} ≠ 0, det {B} ∈ QNow det {AB}= det {A}…

Answered: Let (G, ') denote the set of all 2 x 2…

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter3: Groups

Section3.2: Properties Of Group Elements

Problem 2E: Prove part c of Theorem 3.4. Theorem 3.4: Properties of Group Elements Let G be a group with respect...

See similar textbooks

Similar questions

4. Prove that the special linear group is a normal subgroup of the general linear group .
25. Prove or disprove that every group of order is abelian.
Show that every subgroup of an abelian group is normal.
Prove or disprove that the set of all diagonal matrices in Mn() forms a group with respect to addition.
Prove part c of Theorem 3.4. Theorem 3.4: Properties of Group Elements Let G be a group with respect to a binary operation that is written as multiplication. The identity element e in G is unique. For each xG, the inverse x1 in G is unique. For each xG,(x1)1=x. Reverse order law: For any x and y in G, (xy)1=y1x1. Cancellation laws: If a,x, and y are in G, then either of the equations ax=ay or xa=ya implies that x=y.
Find two groups of order 6 that are not isomorphic.
Let G be a group with center Z(G)=C. Prove that if G/C is cyclic, then G is abelian.
27. a. Show that a cyclic group of order has a cyclic group of order as a homomorphic image. b. Show that a cyclic group of order has a cyclic group of order as a homomorphic image.
Prove that the Cartesian product 24 is an abelian group with respect to the binary operation of addition as defined in Example 11. (Sec. 3.4,27b, Sec. 5.1,53,) Example 11. Consider the additive groups 2 and 4. To avoid any unnecessary confusion we write [ a ]2 and [ a ]4 to designate elements in 2 and 4, respectively. The Cartesian product of 2 and 4 can be expressed as 24={ ([ a ]2,[ b ]4)[ a ]22,[ b ]44 } Sec. 3.4,27b 27. Prove or disprove that each of the following groups with addition as defined in Exercises 52 of section 3.1 is cyclic. a. 23 b. 24 Sec. 5.1,53 53. Rework Exercise 52 with the direct sum 24.
15. Prove that if for all in the group , then is abelian.
Prove or disprove that H={ hGh1=h } is a subgroup of the group G if G is abelian.
Exercises 35. Prove that any two groups of order are isomorphic.

SEE MORE QUESTIONS

Recommended textbooks for you

Elements Of Modern Algebra

Algebra

ISBN:

9781285463230

Author:

Gilbert, Linda, Jimmie

Publisher:

Cengage Learning,

Elements Of Modern Algebra

Algebra

ISBN:

9781285463230

Author:

Gilbert, Linda, Jimmie

Publisher:

Cengage Learning,

SEE MORE TEXTBOOKS