(2) Isomorphic Binary Structures and Subgroups

Question 1

Proving an Isomorphism

Answer 1

Find a function ϕ

• prove ϕ is one-to-one
• onto
• preserves operations

Question 2

If there is an isomorphism from to then G and G’ are said to be ___.

Answer 2

isomorphic

Question 3

Let ϕ be an isomorphism from to . A ____ corresponds to a ______ of G to G’ and vice versa.

Answer 3

Bijection, renaming of elements

Question 4

Let ϕ be an isomorphism from to . ϕ is operation preserving assures that the group are ____. Hence, they have the same table structure.

Answer 4

structurally-alike

Question 5

Why does an isomorphism define an equivalence relation on a set of groups?

Answer 5

https://proofwiki.org/wiki/Isomorphism_is_Equivalence_Relation

Question 6

Let ϕ be an isomorphism from to . If e is the identity of G, then the identity of G’ is ____.

Answer 6

ϕ(e)

Question 7

Group of order 1 up to isomorphism

Answer 7

{e}

Question 8

Group of order 2 up to isomorphism

Answer 8

{e,a}, here the inverse of a is itself

Question 9

T/F. Axiom G3: The identity of G should appear in each row should appear at most once in each row and in each column.

Answer 9

F. Exactly once in each column and in row.

Question 10

T/F. Each element appears exactly once in each column and row.

Answer 10

T

Question 11

There is only one group of order 3 and is ABELIAN. What is it?

Answer 11

{e,a,b} where a and b are inverses of each other

Question 12

Example of groups of order 3 up to isomorphism

Answer 12

Zsub3 under addition modulo 3 and Usub3 under multiplication for C

Question 13

What are the non-isomorphic groups of order 4?

Answer 13

Zsub4 or V

Question 14

What is the group denoted by V?

Answer 14

Klein 4-group

Question 15

Characteristic of V

Answer 15

four elements,
all elements are self inverse
any two produce the third

Question 16

Characteristic of Zsub4

Answer 16

four elements
one element is self inverse
and other two produces the self inverse element

Question 17

H≤G iff

Answer 17

G is a group under * and H≠∅ and H is a subset of G

Question 18

non empty set H is a proper subgroup of a group G iff

Answer 18

H≠G and H is a subset of G

Question 19

non empty set H is an improper subgroup of a group G iff

Answer 19

H=G

Question 20

Define trivial subgroup

Answer 20

{e}

Question 21

If H ≤ G and H ≠ {e} then H is a ______ subgroup

Answer 21

non-trivial

Question 22

T/F. If H is a subgroup of G then the identity of H is different from the identity of G but their inverse elements are the same.

Answer 22

F. Both the identity and inverse elements of H are in G.

Question 23

State the One-Step Subgroup Test

Answer 23

Let G be a group and H a non-empty subgroup of G. Then H is always a subgroup of G iff for all a,b ∈ G, ab^-1 ∈ H.

Question 24

Proving using One-Step Subgroup Test

Answer 24

Let G be a group.

• H is non-empty
• closure
• identity
• inverse

Question 25

Define Special Linear Group of Degree Two

Answer 25

SL(2,R)={[matrix] | a,b,c,d ∈ R and ad-bc = 1}

Question 26

What is the inverse of this matrix
e f
g h

Answer 26

1 h -f
/ [ ]
eh-fg -g e