Give an example of a group G and subgroup H ≤ G such that ♬ is not normal in G.

Q: (h, k) E H x K. What is the order Let H and K be finite groups and let G = H x K. Let g of g?

A: To calculate the order of the element (h,k) in the direct product of groups

Q: Let N, K be two normal subgroups of a group G. Show that NK is a normal subgroup of G.

A: Given that N and K be two normal subgroups of a group G Since N and K are normal subgroup of G…

Q: Let H be a normal subgroup of G and K a subgroup of G that containsH. Prove that K is normal in G if…

A: Assume that H is a normal subgroup of G, and that K is a subgroup of G containing H. Then, we need…

Q: Prove that if H is a normal subgroup of G st H and H/G are finitely so is G.

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Q: Show that the Euclidean group E(n) is a group under function composition.

A: The Euclidean group E(n) consists of all the rigid transformations of n-dimensional Euclidean space.…

Q: Prove that if H is cyclic, then ø[H] is also cyclic.

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Q: If H and K are subgroups of G and g belongs to G, show that g(H N K) = gH n gK.

A: Given: H and K are subgroups of G and g.To prove: gH∩K=gH∩gK

Q: Show that the center Z(G) is a normal subgroup of the group G. Please explain in details and show…

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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 the order of group G =8, show that G must have an element of order 2.

A: Let G be a group and O(G)=8 Also Let a∈G be an arbitrary element other than identity. Then the…

Q: Let G, and G, be two groups. Let H and H, be normal subgroups of G G, respectively then @ H, x H, 4G…

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Q: Let G be a group and H, K≤ G normal subgroups of G. Prove HnK≤G.

A: Given that  is a group and  are normal subgroups of .We need to prove that  is a normal subgroup.We…

Q: Let G be a group and H a normal subgroup of G. Show that if x,y EG Such that xyEH then 'yx€H-

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Q: Let M and N be normal subgroups of G. Show that MN is also a normal subgroup of G

A: It is given that M and N are normal subgroups of G. implies that,

Q: d) G=(Q-(0}, ); H=Q+ e) G=(Zg, ); H=(0,2,4} f) G=the set of 2-tuples of real numbers (a,b) under…

A: In group theory, a branch of mathematics, given a group G under a binary operation ∗, a subset H of…

Q: Let H be a subgroup of G and let K=⋂φ∈Aut(G)φ(H). Show that K is characteristic in G

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Q: 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.

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Q: Let H and K be normal subgroups of a group G such at HCK, show that K/H is a normal subgroup of G/H.

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Q: Write U(32) as the internal direct product of two proper subgroups.

A: Given:         U32 We use the theorem namely Ut≈Usst   because Usst is a subgroup of Ust

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: Show that every subgroup H of the group G of index two is normal.

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Q: (4) Let G be a group. Prove that Z(G) is a normal subgroup of G.

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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: 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: If H₁ and H₂ be two subgroups of group (G,*), and if H₂ is normal in (G,*) then H₂H₂ is normal in…

A: When a non-empty subset of a group follows all the group axioms under the same binary operation, the…

Q: 2. Let G = Z₂ x Z₁. (a) Find subgroups H and K of G such that HK but G/H/G/K. (b) Find subgroups A…

A: (a) Here, G = Z2 × Z4                = { (0, 0), (0, 1), (0, 2), (0, 3), (1, 0), (1, 1), (1, 2), (1,…

Q: disprove: If G is a group and H, K < G are two subgroups, then (HK)≤ G.

A: