Let G be a group. Prove that Z(G) 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: Recall that the center of a group G is the set {x € G | xg = gx for all g e G}. Prove that he center…

A:

Q: . Let H be a subgroup of R*, the group of nonzero real numbers un- der multiplication. If R* C H C…

A: H be a subgroup of R*, the group of nonzero real numbers under multiplication. R+⊆ H ⊆ R*. To prove:…

Q: Let G be an Abelian group and H = {x E G | |x| is odd}. Prove thatH is a subgroup of G.

A: Given: To prove H is a subgroup of G.

Q: 2. Let G be a group. Prove or disprove that z= { XE G: xg = gx for all ge G} is a Subgroup of G.

A:

Q: Let f:G-G be a group homomorphism then H = {a € G:f(a) = a} is subgroup O True False

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 H and K be finite subgroups of a group G and a E G. Then prove that |HaK| = |H||K| /|HnaKa-|.

A: Given that H and K are the finite subgroups of a group G and also an element a such that a∈G Here,…

Q: Let G be a group of order 24. Show that G is solvable

A: Let G be a group of order 24. Show that G is solvable

Q: Let G be a group and g E G. Show that Ø:Z→G by Ø(n) = g" is a homomorphism and isomorphism.

A:

Q: Show that if H and K are subgroups of a group G, then their intersection H ∩ K is also a subgroup of…

A: Subgroup Test           A subset H C G of the group G will be a subgroup if it satisfies the…

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: Show that if H is any group and h is an element of H, with h" = 1, then there is a unique…

A: Given that H is a group and h ∈H  Now,we define a mapping f:Z→H  such that   f(n) = hn for n∈Z For…

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: 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: Let ø be a homomorphism from a group G to a group H. Let K be a subgroup of H. Prove that ø (K) = {…

A:

Q: Let G be a group and H be a normal subgroup of G. If H and G/H are solvable then so is G.

A: Given that Let G be a group and H be a normal subgroup of G. If H and G/H are solvable then so is G.

Q: Let G be a group and a e G. Prove that C(a) is a subgroup of G. Furthermore, prove that Z(G) = NaeG…

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Q: Let G be a group, and let a E G.  Prove that C(a) = C(a-1).

A: Given: Let G be a group and let a∈G. then we will prove C(a)=C(a-1) If C(a) be the centralizer of a…

Q: Let a e G. Prove that $(a") = ¢(a)" for all n e Z.

A:

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: 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: Assume that G is a group such that for all x E G, * x = e. Prove that G is an abelian group.

A: Here we have to prove that G is an abelian group.

Q: Let G be any group with the identity element e. With using the Group Homomorphism Fundamental…

A:

Q: Let G be a group, and let xeG. How are o(x) and o(x) related? Prove your assertion

A: According to the given conditions:

Q: Let G be a group and let Z(G) be the center of G. Then the factor group G/Z(G) is isomorphic to the…

A: We have to check G/Z(G) is isomorphic to group of all inner automorphism of G or not. Where, Z(G) is…

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

A:

Q: Let p : G → G' be a group homomorphism. (a) If H < G, prove that 4(H) is a subgroup of G' (b) If H <…

A:

Q: Let φ : G → H be a group homomorphism. (a) Prove that Ker(φ) is a normal subgroup of G. (a) Prove…

A: To discuss normality of kernel and image under group homomorphisms,

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: Let F and H be subgroups of group G and let FCH. Prove (G : F) = (G : H)(H : F).

A: It is given that F and H are subgroups of G and F⊂H.

Q: Let G be a group. Prove that Z(G) < G.

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: Let H and K be subgroups of a group G and assume |G : H| = +0. Show that |K Kn H |G HI if and only…

A: Given:

Q: Let x belong to a group. If x2e while x : x + e and x + e. What can we say about the order of x? =…

A:

Q: Let G be a non-trivial group. Prove that Aut(G) × Aut(G) is Aut(G x G). a proper subgroup of

A:

Q: If A is an abelian group with A <G and B is any subgroup of G, prove that ANB < AB.

A:

Q: Let G be a group and D = {(x, x) | x E G}. Prove D is a subgroup of G.

A:

Q: Let o be an automorphism of a group G. Prove that H = {x E G | $(x) = x} is a subgroup of G.

A: One step subgroup test: Let G,∘ be a group, then H,∘ is a subgroup of G,∘ if and only if, 1) H is…

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

A:

Q: Let G and H be groups, and let ø:G-> H be a group homomorphism. For xeG, prove that )x).

A: Given:G and H be the groups.