If N is a normal subgroup of a group G, and if every member of N and G/N have a finite order, prove that every member of G has a finite order.

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

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Q: Let G be a finite group, let H be a subgroup of G and let N be a normal subgroup of G. Prove that if…

A: Given that, Let G be a finite group, let H be a subgroup of G and let N be a normal subgroup of G.

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

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Q: 5. Let p and q be two prime numbers, and let G be a group of order pq. Show that every proper…

A: We have to prove that: Every proper subgroup of G is cyclic. Where order of G is pq and p , q are…

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

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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 finite group and let N be a normal subgroup of G such that |N| = n is relatively prime to…

A: Let N be a normal subgroup of a finite group G, and H be any subgroup of G. |N| = n  let |G/N| = p,…

Q: Let G be a finite group, p prime, and Ha Sylow p-group. Prove that H is normal in G if and only if H…

A: Given, G is a finite group and H is a sylow p-group o(G)=n, where n is a finite number. p/o(G) from…

Q: Suppose H is a distant and normal subgroup of a group G. Prove that each subgroup of H is a normal…

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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…

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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 finite group and let primes p and q ≠ p divide |G|. Prove that if G has precisely one…

A: Given that G be a finite group. Consider G be a finite group and consider prime p and q such that…

Q: Suppose S is a nonempty subset of a group G.(a) Prove that if S is finite and closed under the…

A: (a)Suppose S is a non-empty subset of a group G. then we have to prove that if S is finite and…

Q: Let G be a finite p-group of order p". Show that for all 0<kSn, there is a subgroup order p and each…

A: Given: Let G be a finite p-group of order pn. We have to prove for all 0≤k≤n there is a subgroup of…

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

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

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Q: If N is a subgroup of an Abelian group, prove that is Abelian. N |

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

A: this is false because this is need not be true because Z4×Z6 Is not cyclic but have

Q: Let G be a group of order 90. show that G has at most one subgroup of order 45

A: Given: G be a group of order 90

Q: Suppose that G is a group of order 105 with the property that G hasexactly one subgroup for each…

A: Given G is a group of order 105 .

Q: If a simple group G has a subgroup K that is a normal subgroup oftwo distinct maximal subgroups,…

A: Here given G is simple group and K is a normal subgroup of G. Then use the definition of simple…

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: * Let H be a proper subgroup of the group (Z/n)*. Prove that there are infinitely many prime numbers…

A: Given H is subgroup of Z/n* Assume there exist q1, q2 with 0≤q1,q2≤1 and q1≠q2 such that q1+Z=q2+Z…

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: 28) Let N be a finite subgroup of a group G and assume N = for some subset S of G. Prove that an…

A: Given that N is a finite subgroup of a group G and suppose N=S for some subset S of G To prove that…

Q: If H is a subgroup of a group G such that (aH)(Hb) for any a, b eG is either a left or a right coset…

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

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Q: If N is a normal subgroup of G and G/N is abelian. Then G is also abelian. Select one: True False

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Q: G is a finite group of order IGI =pqr with p< q <r prime

A: Solution

Q: If G is a cyclic group, prove for subgroup N that G is a cyclic N

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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: Prove that if G is an abelian group of order n and s is an integer that divides n, then G has a…

A: G is an abelian group of order n ; And, s is an integer that divides n;

Q: 6. Let G be a group of order p², where p is a prime. Show that G must have a subgroup of order p.

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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…

Equations and Inequations

Equations and inequalities describe the relationship between two mathematical expressions.

Linear Functions

A linear function can just be a constant, or it can be the constant multiplied with the variable like x or y. If the variables are of the form, x2, x1/2 or y2 it is not linear. The exponent over the variables should always be 1.