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

Q: Let G be a finite group of order n. Let g be an element of G. Prove that gn=e.

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Q: Exercise 8.3. (a) If H₁ and H₂ are subgroups of groups G₁ and G₂, respectively, prove that H₁ÐH₂ ≤…

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Q: All groups of order three are isomorphic.

A: All groups of order three are isomorphic.

Q: Prove that the group G = [a, b

A: Given, the group G=a, b with the defining set of relations…

Q: c) Show that if G is a group of order 100, then G has at most one subgroup of order 25.

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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: Give an example of a p-group of order 9.

A: Given, Give an example of a p-group of order 9.

Q: that a and xax^-1 have same order for all x belongs G.

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Q: V Let G be a group. Show that for all a, b in G, (a*b) =b*a

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Q: Let h: G G be a group homomorphism, and gEG is an element of order 35. Then the possible order of…

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

A: Given :              A group of order 4  Corollary :                    The order of an element of a…

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 group. Show that for all a,b,c G, (acb-1)-1 = bc-la.

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Q: For a group G and a fixed element a ∈ G, define the subset C(a) to be the set of all elements in G…

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Q: Let G be a cyclic group of order n. Let m < n be a positive integer. How many subgroups of order m…

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Q: If G is a finite group, H ≤ G, the order of H divides the order of G: | H | / | G | Prove

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Q: Let a be an element in group G, order (a)=15, then the order a10 = 3, where a10 E G

A: Use the formula of o(a^r) , if o(a) is given.

Q: Let (G, *) and (H, *) be groups. The direct product of G and H is the set G × H equipped with…

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Q: suitte (b) Given two groups (G,) and (H, *). Suppose that is a homomorphism of G onto H. For BH and…

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Q: Exercise 7.15. Let G be a finite group, g € G, and N≤ G. Prove that the order of gN in G/N must…

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

A: Let G be a group. Let a be an element of order 8 of group G. That is, a8=e  where e is an identity…

Q: Let a,b be elements of a group G .Assume that a has order 5 and  a^3b=ba^3. prove that ab=ba.

A: Let a,b be elements of a group G .Assume that a has order 5 and a^3b=ba^3. prove that ab=ba.

Q: Let G, H be cyclic groups of order 2 and 3, respectively. Find the orders of all elements of G x H

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Q: order 8 of a group G, and =

A: Given that order of a is 8 .Then  a8=e Rearrange  a little bit , we can have a42=e Hence order of…

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 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: Q\ Let (G,+) be a group such that G={(a,b): a,b ER}. Is ({(0,a): aER} ,+) sub group of (G,+).

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Q: Let G be a group of finite order n. Prove that an = e for all a in G.

A: Let G be a group of finite order n with identity e.           Since G is of finite order…

Q: Determine whether the set G is a group under the operation * G= integers Z ; a*b= a-b

A: According to the given information it is required to check whether the given set G is a group or…

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

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Q: Let G be the set (te, ±a, ±b, ±c) where [ 1 10 Show that G form a Group under multiplication.

A: Group: If a set (finite or infinite) together with a binary operation satisfies the four basic…

Q: Let G1 and G2 be groups. Prove that G1 × G2 is isomorphic to G2 × G1.

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Q: Let a be an element in group G, order (a)=15, then the order a10 = 3, where a10 e G

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Q: Let G be a cyclic group of order 33. List the orders of all its elements and how many elements have…

A: Here, G is a cyclic group of order 33.  We have to find the orders of all its elements and the…

Q: Let (G,*) be a group such that a² = e for all a E G. Show that G is commutative.

A: A detailed solution is given below.

Q: Let G be a finite group of order 4 containing no element of order 4. Explain why every nonidentity…

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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: Let G be a group having two finite subgroups H and K such that gcd(|H.K) 1. Show that HOK={e}.…

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Q: Let G be a group of order p™ where p is a prime number and m is a positive integer. Show that G…

A: given G be a group of order pm where p is a prime number and m is a positive integer to show G…

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

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Q: Let G be a group, a E G. Prove that a=a + a < 2

A: Concept:

Q: Determine whether the set G is a group under the operation * G={n integer|n is odd}; a*b=a+b

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Q: F. Let a e G where G is a group. What shall you show to prove that a= q?

A: Solution: Given G is a group and a∈G is an element. Here a-1=q

Q: Let G be a group with order n, with n> 2. Prove that G has an element of prime order.

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Q: Let a, b be elements of a group G. Assume that a has order 5 and a³b = ba³. Prove that ab = ba.

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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: G be the external direct product of groups G, G2.. H, = {4,e2.e*, e..e,x, e G,}

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