Answer to Question #123924 in Abstract Algebra for Damilani

Question

1. Build up the operation tables for group G with orders 1, 2, 3 and 4 using the elements a, b, c, and e as the identity element in an appropriate way.
2. i. State the Lagrange’s theorem of group theory.
ii. For a subgroup H of a group G, prove the Lagrange’s theorem.
iii. Discuss whether a group H with order 6 can be a subgroup of a group with order 13 or not. Clearly state the reasons.

Answer 1

Operation table

Table 1

*eabc

eeabc

aaecb

bbcae

ccbea

Table 2

*eabc

eeabc

aaecb

bbcea

ccbea

Table 3

*eabc

eeabc

aabce

bbcea

cceab

Table 4

*eabc

eeabc

aaceb

bbeca

ccbea

Second part state lagrange theorem

If G is a finite group and H is a subgroup of a G then |H| divides |G|

and [G:H]=|G|/|H|

Third part

For any g belongs to G,h belongs to H;

(gh)² belongs H and g^-2belongs H

Since H is a subgroup, h^-2 g^-2belongs H and so (gh)² h^-1g^-2belongs H.

This gives that gh gh h^-1g^-2belongs H

I.e. ghg^-1belongs H.

Hence H is a normal subgroup of G.

non empty subset H of a group G is a subgroup of G if and only if

1) a belongs H,b belongs H⟹a*b∈ H

2)a∈ H⟹ a^-1∈ H

For example G={-5,-4-3,-2,-1,0,1,2,3,4,5,6,7,}

Order 13

H={-3,-2,-1,0,1,2}

ORDER 6

PROERTY FOLLOWS

1)1*2∈ H

2)1^-1∈ H

So H is a subgroup of G