(6) Cartesian Product to Finitely Generated Abelian Groups

Question 1

Cartesian Product of Ssubi

Answer 1

S1 x S2 x … x Si with 1 ≤ i ≤ n

Question 2

The Cartesian Product of Si from 1 to n is a group under what operation? The group is called?

Answer 2

componentwise multiplication

external direct product

Question 3

The symbol for direct product

Answer 3

https://en.wikipedia.org/wiki/Direct_product

Question 4

Define c as the LCM of a and b

Answer 4

a|c and b|c

for all n ∈ ℤ s.t. a|n and b|n c|n

Question 5

Let G1, G2, G3,..,Gn be finite groups and (a1,a2,a3,..,an)) ∈ their direct product, then what is the order of (a1,a2,a3,…,an) in the direct product?

Answer 5

LCM(|a1|,|a2|,…,|an|)

Question 6

ℤm x ℤn ≅ ℤmn iff m and n are

Answer 6

relatively prime

Question 7

Is ℤ10 x Z10 ≅ Z100?

Answer 7

No. 10 is not rel. prime to itself

Question 8

G1 x G2 ≅ G1 x G3 only if ?

Answer 8

G2 ≅ G3

Question 9

Let G be a group and X ⊆ G. What is the smallest subgroup of G containing all the elements of X?

Answer 9

, subgroup generated by X

Question 10

Let a ∈ G with id: e.
=?
=?
=?
=?

Answer 10

{e}
{e}
<a>
G</a>

Question 11

is the set containing?

Answer 11

product of integral powers of elements in X

Question 12

T/F/ is the intersection of all subgroups of G containing X.

Answer 12

T

Question 13

G is finitely generated when

Answer 13

there exist X ⊆ G such that =G and X is finite

Question 14

If G=<a> then G is finitely generated. Why?</a>

Answer 14

<a>=</a>

Question 15

Every finite group is finitely generated. Why?

Answer 15

G=

Question 16

T/F. If a group is noncyclic then it can not be finitely generated

Answer 16

F. e.g. D4=

Question 17

Define the Quarternion Group Q8.

Answer 17

Q8={1,-1,i,-i,j-,j,k,-k}
ijkij
a ^ 2 = 1 where a ∈ {i,j,k}
-a= -1 times a

Question 18

If G1,G2,..,Gn are CYCLIC groups then their direct product is finitely generated with generating set

Answer 18

{(a1,e2,..,en), (e1,a2,..,en), (e1,e2,..,an)} where ei is the id of Gi and Gi= for 1 ≤ i ≤ n.