Let \(G\) be an abelian group and \(H,K\) two finite subgroups of \(G\) of orders \(m\) and \(n\), respectively. Prove that \(G\) has a subgroup of order \(LCM(m,n)\) .

(a) Show that the center of a group \(G\) is a characteristic subgroup of \(G\).

(b) Prove that characteristic subgroups are normal. Give an example of a normal subgroup which is not characteristic.

(c) Show that if \(H\) is the only subgroup of order \(n\) in \(G\), or if \(H\) is the only subgroup of index \(k\) in \(G\), then \(H\) is characteristic in \(G\).

Prove that there are no simple groups of orders 42, 56, 96, 108, 200, 1986.

Let \(G\) be a group of order \(1575\). Prove that if \(H\) is a normal subgroup of order 9 in \(G\), then \(H \le Z(G)\).

Show that if \(|G| =pqr\) , where \(p,q,r\) are distinct primes, then \(G\) is not simple.

If a group of order \(p^n\) , where \(p\) is a prime, contains exactly one subgroup each of orders \(p,p^2, \cdots ,p^{n-1}\) , prove that it is cyclic.

Let \(G\) be a group. Prove that \(|G/Z(G)| \neq 77\) .

If \(G\) is a group with \(|G|=p^n\) , where \(p\) is a prime, and if \(0 \le k \le n\) , show that \(G\) contains a normal subgroup of order \(p^k\) .

Let \(H\) be a proper subgroup of a finite p-group \(G\) . If \(H=p^s\) , then prove that there is a subgroup of \(G\) order \(p^{s+1}\) containing \(H\) .

Prove that a finite p-group cannot be simple unless it has order p

Let G be a finite group and p a prime. Show that a normal p-subgroup of G is contained in every Sylow p-subgroup of G.

If \(P\) is a normal Sylow p-subgroup of \(G\) and \(H \le G\), prove that \(P \cap H\) is a unique Sylow p-subgroup of \(H\).

Let \(H\) be a subgroup of a finite group \(G\) and \(p\) a prime. Prove that \(|Syl_p(H)\ \le |Syl_p(G)|\) .

If \(G\) is a finite group and \(p \in Syl_p(G)\), show that \(N_G(N_G(P))=N_G(P)\) .

Let \(G\) be a finite group and \(H \le G\). Suppose that \(P \in Syl_p(H)\) . If \(N_G(P) \subseteq H\), show that \(P \in Syl_p(G)\) .

Let \(G\) be a finite group and \(P \in Syl_p(G)\) , and let \(H\) be a subgroup of \(G\) containing \(N_G(P)\) . Prove that \(N_G(H) =H\) .

Show that, if \(\sigma \in S_n\) is a cycle of length \(r\), then \(o(\sigma) =r\) .

Let \(\sigma \in S_n\) be a product of disjoint cycles \(\alpha_1,\alpha_2, \cdots ,\alpha_r\) of lengths \(n_1,n_2, \cdots ,n_r,\) respectively. Prove that \(o(\alpha) = LCM(n_1,n_2, \cdots ,n_r)\).

All Questions

Physics

Chemistry

Mathematics

English

Organic Chemistry

Inorganic Chemistry

Physical Chemistry

Algebra

Geometry

Evaluate :

∫2xcos(x2 – 5).dx

If the sum of 14 terms of an A. P is 1050 its first term is 10 find 20th term ?

Diagonals of a quadrilateral 30 cm long and the perpendicular drawn from the opposite vertices are 5.6 and 7.4 cm find the area of the quadrilateral.with steps ?

Find the sum of the integers between 10 and 30 including 10 and 30 which are not divisible by 3.

A cyclic quadrilateral pqrs are where PS equals to PQ and angle SPQ is 70 degree find angle psq