Algebra
Algebra in GATE MA tests group theory (Lagrange, normal subgroups, Sylow) and ring theory (ideals, polynomial rings, fields). Problems are often short proofs or classification of small-order groups.
Core rules
- Lagrange: |H| divides |G| for a subgroup H. The converse is false in general.
- A subgroup H is normal iff gHg^{−1}=H for all g; quotient G/H is a group exactly then.
- First isomorphism theorem: G/ker φ ≅ im φ.
- Sylow: n_p ≡1 mod p and n_p divides |G|/p^k; n_p=1 implies a normal Sylow p-subgroup.
- In a PID every ideal is principal; Euclidean domain ⇒ PID ⇒ UFD. Eisenstein's criterion gives irreducibility over Q.
- A group of order p² is abelian.
Worked example
Show every group G of order 15 is cyclic. Sylow: n_3≡1 mod 3 and n_3|5, so n_3=1; n_5≡1 mod 5 and n_5|3, so n_5=1. Thus the Sylow 3- and 5-subgroups are normal. Their product has order 15 and is abelian because the subgroups intersect trivially and are normal. A group of order pq with p<q and p∤(q−1) is cyclic, so G≅Z_15.
Traps
- Normality is not transitive; H normal in K and K normal in G does not make H normal in G.
- Sylow counts must satisfy both congruence and divisibility; missing one gives the wrong n_p.
- An ideal is not just a subring; it must absorb multiplication by all ring elements.
- A polynomial can be irreducible over Q but reducible over R or C; specify the field.