Let G be a group and let A1, A2, B1, B2 be the proper nontrivial subgroups of G. Suppose that A1×A2 is isomorphic to B1×B2. Is it true that A1 must be isomorphic to B1 or B2

Q: ) Let G be a finite group , IGI=ps. p prime Prove that G cannot have two distinct and sep. subgroups…

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

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Q: Let (G,*) be an a belian group, if (H,) and (K,*) are subgroup of (G,*) then (H * K,*) is a subgroup…

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Q: . Let N be a finite group and let H be a subgroup of N. If [H| is odd and [N:H] = 2, prove that the…

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

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

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Q: If H is a cyclic subgroup of a group G then G is necessarily cyclic * True False

A: let H be a cyclic subgroup of a group G.

Q: Assume that (G, ) is a group and that (H, ) and (K, ) are subgroups of (G,*). Prove that (HnK,*) is…

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Q: Suppose that G is a group and |G| = pnm, where p is prime andp >m. Prove that a Sylow p-subgroup…

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

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Q: If G is a group and a1, a2,…, an shows that a1 * a2 *… * an is unique, regardless of the order in…

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Q: If a is an element of order 8 of a group G, and

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Q: Let N be a finite group and let H be a subgroup of N. If |H| is odd and [N:H] = 2, prove that the…

A: According to the given information, Let N be a finite group and H be a subgroup of N. H is odd and…

Q: Let G be a group, H,K ≤ G such that H=, K=for some a,b∈G. That is H and K are cyclic subgroups of G.…

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Q: It is not possible that, for a group G and H and K are nomal subgroups of G, H is isomorphic to K…

A: Let G be a group and H and K are normal subgroups of G

Q: Show that a group of order 4 either is cyclic or is isomorphic to the Klein four group {e, a, b, a…

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Q: let G be a group of order p^2 where p is prime. Show that every subgroup of G is either cyclic or…

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Q: Let Z denote the group of integers under addition. Is every subgroup of Z cyclic? Why? Describe all…

A: Yes , every subgroup of z is cyclic

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

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Q: Let (G,*) be an a belian group, if (H,*) and (K,*) are subgroup of (G,*) then (H * K,*) is a…

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Q: Let H  and K  be subgroups of a group G with operation * .  Prove that HK  .is closed  under the…

A: Given information: H  and K  be subgroups of a group G with operation * To prove that HK is a closed…

Q: Suppose G is a group of order 48, g € G, and g" = €. Prove that g = ɛ.

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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…

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

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Q: Show that if aH=H then a belongs to H. H is a subgroup of a group G and a is an element of G

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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: Given orders of subgroup 10 18 30

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

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Q: Prove that any two Sylow p-subgroups of a group G are conjugate.

A: To prove that any two sylowp-subgroup of a group G are conjugate

Q: For any group G, GIZ(G) is isomorphic to Inn(G)

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Q: Let H and K be subgroups of a group G. Prove that HNK is a subgroup of G.

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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: Show that if G is a finite group with identity e and with an even number of elements, then there is…

A: Given,If G is a finite group with identity e and with an even number of elements,then there is a ≠e…

Q: If G is a cyclic group of order n, prove that for every element a in G,an = e.

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Q: Let G be a finite group of order 125 with the identity element e and assume that G contains an…

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Q: Suppose the o and y are isomorphisms of some group G to the same group. Prove that H = {g E G| $(g)…

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Q: A subset of H of a group G is a subgroup of G if the operation on G makes H into a group. Prove that…

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Q: Let G be a group and g E G. Prove that if H is a Sylow p-group of G, then so is gHg-1

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Q: Suppose that G is a group and |G| = pnm, where p is prime and p &gt; m. Prove that a Sylow…

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Q: If H is the subgroup of group G where G is the additive group of integers and H = {6x | x is the…

A: Let H is a subgroup of order 6 . Take H=6Z where Z is integers.

Q: Let a be an element of a group G such that Ord(a) = 32. If H is a normal subgroup of G, then Ord(aH)…

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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: Let (G1, ) and (G2, *) be two groups and p: G1 G2 be an isomorphism. The

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Q: Prove that ifH and K are subgroups of a group G with operation *, Question 8. then HNK is a subgroup…

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Q: Let a and b be nonidentity elements of different orders in a group Gof order 155. Prove that the…

A: Given that Let a and b be nonidentity elements of different orders in a group G of order 155. Note…

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

A: According to the given information it is required to prove that every group of order 375 has a…