Prove that C(a) is a subgroup of G.

Q: 3. Prove that a subset H of a finite group G is a subgroup of G if and only if a. His nonempty, and…

A: We have to prove given property:

Q: Prove that, if H is a subgroup of a cyclic group G, then the quotient group G/H is also cyclic.

A:

Q: Prove that a group of order 375 has a subgroup of order 15.

A:

Q: Prove that if N is a normal subgroup of G, and H is any subgroup of G, then H ∩ N is a normal…

A: To Prove If N is a normal subgroup of G, and H is any subgroup of G, then H ∩ N is a normal subgroup…

Q: 2) Let H be a normal subgroup of G. If| H|-2. Prove that H is contained in the center Z(G) of G.

A:

Q: Find a proper subgroup of the group of integers Z under addition and prove that this subgroup is…

A: Consider 2 ℤ = {an / n∈ ℤ }This is a subgroup of ℤ .claim : 2ℤ is isomorphic to ℤ.…

Q: Let ø be a homomorphism from a group G to a group G'. Prove that ker øis a subgroup of G.

A:

Q: 6. If G is a group and H is a subgroup of index 2 in G; then prove that H is a normal subgroup of G:

A: I have proved the definition of normal subgroup

Q: (a) Prove that if K is a subgroup of G and L is a subgroup of H, then K x L is a subgroup of G x H.

A: The detailed solution of (a) is as follows below:

Q: let G be a group and H a subgroup of G. prove that for any element gEG holds that gH=H if and only…

A: We can solve the given question as follows:

Q: (a) of G'. Show that if y :G → G' is a group homomorphism then Im(y) is a subgroup

A: According to the given information, For part (a) it is required to show that:

Q: Prove that the intersection of two subgroups of a group G is a subgroup of G.

A: We will prove the statement.

Q: Let G be a group with |G| = pq, where p and q are prime. Prove that every proper subgroup of G is…

A:

Q: 32. If H and K are subgroups of G, show that Hn K is a subgroup of G. (Can you see that the same…

A: To show:

Q: Suppose G is a group and Z (G) and lnn (G) are the centers and groups of internal deformations of G,…

A: Let G is a group and Z (G) and lnn (G) are the centers and groups of internal deformations of G

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

A:

Q: Find a proper subgroup of the group of integers ? under addition and prove that this subgroup is…

A: Solution: The objective is to find a proper subgroup of the group of integers and to show that this…

Q: H be a subgroup of G.

A: We have to find out the truth value of the given statements. It is given that H is a subgroup of G.…

Q: True or false? The union of any two subgroups of a group G is a subgroup of G.

A:

Q: Let G be a group with |G|=187 then every proper subgroup of G is:

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: 5. Suppose G is a group of order 8. Prove that G must have a subgroup of order 2.

A:

Q: a. Prove or Disprove. If H and K be normal subgroups of a group G and H is isomorphic to K, then G/H…

A:

Q: 1- Prove that if (Q -(0),) is a group, and H = an, m e Z} 1+2m is a subset of Q - {0)}, then prove…

A: A subset H of a group G, · is said to be a subgroup of G, · if for any a,b∈H we have: a·b∈H a-1∈H…

Q: Let G be a group and let H and K subgroup of G. Prove that the intersection H and K is a subgroup of…

A:

Q: Question 2: If p is a homomorphism of group G onto & with kernel K and N is a normal subgroup of G.…

A: Introduction: If there exists a bijective map θ:G→G' for two given groups G and G', then θ is…

Q: If H and K are subgroups of a group G, prove that ANB is a subgroup of G.

A: GIVEN if H and K are the subgroup of a G, prove that  A∩B  is a subgroup of G

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: let H be a normal subgroup of G and let a belong to G . if the element aH has order 3 in the group…

A:

Q: Let G be a group with the order of G = pq, where p and q are prime. Prove that every proper subgroup…

A: Consider the provided question, Let G be a group with the order of G = pq, where p and q are prime.…

Q: Let G be a group of order 24. If H is a subgroup of G, what are all the possible orders of H?

A: Given, o(G)=24 wherre H is a subgroup of G from lagrange's theoram: for any finite order group of G…

Q: Prove that a normal subgroup need not to be a characteristic subgroup.

A:

Q: Let H be a subgroup of G of index 2. Prove that H is a normal sub-group of G.

A: the prove is given below...

Q: 5.22 Let G be a group. Prove that Z(G) is a subgroup 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: Let let G N Subgroup be be of G a a group and normal of finite

A: To prove that H is contained in N, we first prove this: Lemma: Let G be a group.H⊂G. Suppose, x be…

Q: Let G be a group. Prove that Z(G) is a subgroup of G.

A: The set ZG=x∈G|xg=gx,∀g∈G of all elements that commute with every other element of G is called the…

Q: Prove that the intersection of two subgroups is always a subgroup.

A: In this question, we prove the intersection of the two subgroup of G is also the subgroup of G.

Q: If op is a homomorphism of group G onto G with kernel K and Ñ is a normal subgroup of G. N = {x E G|…

A: What is Group Homomorphism: If there exists a bijective map θ:G→G' for two given groups G and G',…

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 any group of order 75 can have at most one subgroup of order 25.

A:

Q: If H is a subgroup of G such that [G : H] = 2, then show that H is a normal subgroup of G.

A: Suppose H≤G such that [G:H] = 2. Thus H has two left cosets (and two right cosets) in G.

Q: Prove that the intersection of two subgroups of G is itself a subgroup of G.

A: Suppose that, H1 and H2 be the sub groups of G. Also, let H=H1∩H2 . Consider one step group test:…

Q: 2) Let (G, *) be a group and H, K be subgroups in G. Prove that subset H * K is a subgroup if and…

A:

Q: Suppose that G is a group that has exactly one non-trivial proper subgroup. Prove that G is cyclic.

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Q: Prove that every group of order 78 has a normal subgroup of order 39.

A:

Q: Prove that if H is normal in G and ø is onto, then ø[H] is normal in G'.

A:

Q: Show that every group G of order n is isomorphic to a subgroup of Sn. (This is also called Caley's…

A: