Algebra 2 Notes 2

Question 1

(2D-N1-2.1) A group - and everything with it inc. Abelian or Commutative. Closed Binary Operation

Answer 1

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Question 2

(2L-N1-2.2) if ⋆ is an associative binary operation on a set S, then for any x1, …, xn ∈ S, any bracketing of x1 ⋆ x2 ⋆ … ⋆ xn yields the same answer.

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Question 3

(2L-N1-2.3) If ⋆ is a binary operation on a set G and e and f are both identity elements, then e = f.

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Question 4

(2L-N2-2.4) Let G be a set, ⋆ be an associative binary operation on G which has an identity element e. Then if both g and h are inverses of an element f ∈ G, then g = h.

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Question 5

(2L-N2-2.5) For any element g of a group G,(i) (g−1)−1 = g, (ii) (g ⋆ h)−1 = h−1 ⋆ g−1

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Question 6

(2D-N2-2.6) Define g2 = gg, g3 = ggg, etc Also define g−n = (g−1)^n(n ∈ N) and g0 = e.

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Question 7

(2L-N2-2.7) For any m,n ∈ Z, usual rules for indices hold

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Question 8

(2P-N2-2.8) i) If f,g,h ∈ G, a group, and fg = fh then g = h (ii)Let G be a group and g1 ∈G. Then g1G={g1x:x∈G} contains each element of G exactly once. In particular if G consists of a finite number of elements g1, …, gn, then the list gg1, gg2, …, ggn is just re-ordering of g1, …, gn.

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Question 9

(2R-N3-) Representing a group with a group table

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Question 10

(2L-N4-2.9) Let X be a set and let S(X) = {f : X −→ X s.t. f is bijective} Let ◦ be composition of functions. Then S(X) is a group under ◦.

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Question 11

(2D-N4-2.10) If X = {1, 2, …n}, the group S(X) is normally denoted Sn and called the symmetric group; elements of Sn are called …Define formally integers mod m

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Question 12

(2T-N4-2.11) (a) For any integer m, Zm under addition forms an abelian group. (b) For any prime p, Z∗p, the set of non-zero elements of Zp under multiplication, forms an abelian group.

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Question 13

(2D-N5-2.12) (i) An isometry of the plane R^2. (ii) If T is any set of points of the plane, then Sym(T) …

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Question 14

(2L-N5-2.13) Sym(T) under composition forms a group.

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Question 15

(2D-N6-2.14) Order of group, element

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Question 16

(2L-N6-2.15) Let G be a group, and g ∈ G with o(g) = n. Then (i)gm =e⇔n divides m. (ii) any power of g is equal to exactly one of the elements of the set {e, g, g2, …, gn−1}.

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Question 17

(2D-N6-2.16) G generated by g. G cyclic.

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Question 18

(2L-N6-2.17) Let G be a finite group of order n. Then G is cyclic if and only if it contains an element of order n.

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Question 19

(2D-N6-2.18) Let G be a cyclic group, generated by g. If o(g) = n then the distinct elements of G are e, g, g2, g3, …, gn−1 and G is … (ii) If o(g) = ∞, …

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Question 20

(2R-N7-) Z under + is isomorphic to C∞

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Question 21

(2D-N7-2.19) H subgroup. Example w/ permutations

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Question 22

(2L-N7-2.20) Let G be a group and H a subset of G. H is a subgroup of G if it forms a group under the same operations as G (i.e. the same multiplication which implies also the same identity element and the same inverses).

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Question 23

(2T-N7-2.21) Let An denote the set of even permutations in Sn. Then An forms a subgroup of Sn, of order n!/2, called the alternating group.

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Question 24

(2T-N8-2.22) Lagrange’s Theorem

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Question 25

(2C-N8-2.25) Let g ∈ G, where G is a finite group. Then o(g) divides |G|.

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Question 26

(2C-N9-2.26) Let G be a group of order p, where p is a prime. Then G is cyclic of order p.

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Question 27

(2T-N9-2.27) Fermat’s Little Theorem.

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Question 28

(2D-N10-2.3) Group Homomorphism

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Question 29

(2D-N10-2.31) Let φ : G −→ H be a group homomorphism. Then the kernel of φ is

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Question 30

(2P-N10-2.32) Let φ : G −→ H be a group homomorphism. Then (i) φ is injective if and only if Ker(φ) = {e}. (ii) φ is surjective if and only if Im(φ) = H.

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Question 31

(2D-N11-2.33) Let φ : G −→ H be a group homomorphism. If φ is bijective…

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Question 32

(2E-N11-) Some example of isomorphic groups

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