Give an example of a finite group G with two normal subgroups H and K such that G/H = G/K but H 7 K.

Q: If p is a prime, prove that any group G of order 2p has a normal subgroup of order p and a normal…

A: To prove that any group of order 2p has a normal subgroup of order p and a normal subgroup in g

Q: Prove that a group of order n greater than 2 cannot have a subgroupof order n – 1.

A: Given: To Prove: G cannot have a subgroup of order n-1.

Q: Suppose that a group G of order 231 has a normal subgroup N of order 11. Then, G/N is cyclic O False…

A: Given that G is a group of order 231 and N is an normal sub-group of G of order 11. To show: G/N is…

Q: Prove that a simple group of order 60 has a subgroup of order 6 anda subgroup of order 10.

A: If G is the simple group of order 60 That is | G | =60. |G| = 22 (3)(5). By using theorem, For every…

Q: is the smallest order of a group that contains both a subgroup isomorphic to Z12 and Z18?

A:

Q: Let (G,*) be a group of order p, q, where p, q are primes and p < q. Prove that (a). G has only one…

A: It is given that G, * is a group of order p·q where p, q are primes and p&lt;q. Show that G has only…

Q: G. Show that if H and K are subgroups of an abelian group G, then {hk\h E H andke K} is a subgroup…

A:

Q: Q2: Let (G,) be a commutative group, and let the set H consist of all elements of G with finite…

A: Given a group G and a set H of G with the given conditions. We need to show that H is a normal…

Q: Use the fact that a group with order 15 must be cyclic to prove: if a group G has order 60, then the…

A:

Q: Let N be a normal subgroup of a finite group G. Use the theorems ofthis chapter to prove that the…

A:

Q: If K is a normal subgroup of a finite group G and S is a Sylow p-suby

A: Given that if K is a normal subgroup of a finite group  G and S is a Sylow p-subgroup of G. then K∩S…

Q: Suppose that G is a group of order 168. If G has more than oneSylow 7-subgroup, exactly how many…

A:

Q: 9. Prove that if G is a group of order 60 with no non-trivial normal subgroups, then G has no…

A:

Q: 5/ Let G be group of class p9 a Prime Setting that proves that actual Subgroup of G is a cyclie is a

A: We know that every group of prime order is cyclic

Q: Let G be a group of order 100 that has a subgroup H of order 25.Prove that every element of G of…

A:

Q: If H is a normal subgroup of a finite group G and |H| = pk for someprime p, show that H is contained…

A: H is a normal subgroup of a finite group G and |H| = pk for some prime p.

Q: 12. Prove that the intersection of any family of normal subgroups of a group (G, *) is again normal…

A:

Q: Let G be a finite group and H a subgroup of G of order n. If H is the only subgroup of G of order n,…

A: Given, G is a finite group and H is a subgroup of G of order n.

Q: Let G be a finite group. Then G is a p-group if and only if |G| is a power of p. We leouo the

A: Given G is finite group and we have to prove G is a p-Group of and only if |G| is a power of p.

Q: Let G be a cyclic group of order n. Let m < n be a positive integer. How many subgroups of order m…

A:

Q: If a cyclic group T of G is normal in G; then show t subgroup of T is a normal subgroup in G

A: Given: A cyclic group T of G is normal in G.

Q: 17. Show that every group of order (35)° has a normal subgroup of order 125.

A:

Q: A group G has order 4n. where n is odd. Show that G has no subgroup of order 8.

A:

Q: If N is a normal subgroup of order 2 of a group G then show that N CZ(G).

A:

Q: (8) If H1, H2 are 2 subgroups of G, prove that H1 N H2 is also a subgroup of G. If further assume…

A:

Q: If G is a finite group with |G|<180 and G has subgroups of orders 10, 18 ano then the order of G is:…

A:

Q: 6. Prove that if G is a group of order 231 and H€ Syl₁1(G), then H≤ Z(G). n Core

A: Given that,  G is group of order 231 and H∈syl11G. We first claim that there is a unique Sylow…

Q: Prove that if H is a normal subgroup of G of prime index p. (Note G can be finite or infinite…

A: It is given that, H is a normal subgroup of G of prime index p. (Here G can be a finite or infinite…

Q: Let G be an Abelian group and H 5 {x ∊ G | |x| is 1 or even}. Givean example to show that H need not…

A:

Q: (4) Let G be a group and H ≤ G. The subgroup H is normal in its normalizer NG(H), this imply that…

A:

Q: Suppose that G is a finite simple group and contains subgroups Hand K such that |G:H| and |G:K| are…

A: Consider the finite simple group G that has subgroup H and K. |G: H| and |G: K| are relatively…

Q: Exercise 7.16. Prove that if N is a normal subgroup of the finite group G and ged(|N|, |G/N|) = 1,…

A:

Q: If G is a finite group with |G|<180 and G has subgroups of orders 10, 18 and 30 then the order of G…

A:

Q: Prove that if G is a finite group and H is a proper normal subgroupof largest order, then G/H is…

A: Given: G is a finite group and H is a proper normal subgroup of largest order.

Q: Use the fact that a group with order 15 must be cyclic to prove: if a group G has order 60, then the…

A:

Q: H. Show that an intersection of normal subgroups of a group G is again a normal subgroup of G.

A:

Q: 7. Let G be a group, prove that the center Z(G) of a group G is a normal subgroup of G.

A:

Q: Show that if G is a group of order 168 that has a normal subgroup oforder 4, then G has a normal…

A:

Q: Prove that every group of order 675 has a normal Sylow 5-subgroup.

A: Theorem: If number of Sylow p-subgroup of a group G is unique, then that subgroup is normal. Let G…

Q: The Kernal of any group homomorphism is normal subgroup True False

A:

Q: Let G be a group having two finite subgroups H and K such that gcd(|H.K) 1. Show that HOK={e}.…

A:

Q: Prove that if G is a group of order 60 with no non-trivial normal subgroups, then G has no subgroup…

A:

Q: Let G be a finite group and let H be a normal subgroup of G. Provethat the order of the element gH…

A: Given: G be a finite group and H be a normal subgroup of G.

Q: Let let G₁ be A be of Suppose Subgroup index a group and a normal of finite G+₁ that H

A: We know that if G is a group and H is a subgroup of G and x is an element in G of finite order n. If…

Q: Let G be a group with order n, with n> 2. Prove that G has an element of prime order.

A:

Q: Show that if H and K are subgroups of an abelian group G then {hk: h element of H and k element of K…

A:

Q: Let a be an element of a group G such that Ord(a) = 30. If H is a normal subgroup of G, then Ord(aH)…

A:

Q: Suppose that H is a subgroup of Sn of odd order. Prove that H is asubgroup of An.

A: Given: H is a subgroup of Sn of odd order, To prove: H is a subgroup of An,

Q: Let G be a group of order 24. Suppose that G has precisely one subgroup of order 3, and one subgroup…

A: Theorem : If a group G is the internal direct product of subgroups H and K, then G is isomorphic to…