Normal Subgroups vs P-Sylow Subgroups

Question

Let group G= Z_6 × S_3 and H, K be 2 subgroups of G such that |H|=4 and |K|=9

I know |G|=2^2 * 3^2

What is the normality between the groups

(i) H & G

(ii) K & G

I know that a group is normally iffy the no of P-SSG of it is unique.

What does unique here means?

How do we find if H&G and K&G are normal?

Answer 1

Terms:

1. Group = A set with an operation (like addition) that follows certain rules.
2. Subgroup = A smaller group within a larger group.
3. Order of a Group = |G| = The number of elements in the group.
4. Normal Subgroup = A subgroup that fits nicely within the group.
5. P-Sylow Subgroup = A special subgroup related to a prime number p.

Walk-through:

1. Is H Normal in G?
2. H has 4 elements (2²).
3. If there is only one subgroup of order 4 in G, H is normal.
4. Is K Normal in G?
5. K has 9 elements (3²).
6. If there is only one subgroup of order 9 in G, K is normal.

Conclusion:

1. Count the subgroups of order 4 and 9 in G.
2. If there's only one of each, H and K are normal.