Let G be a group and a be an element of this group then necessarily: * Jal<|G| O lal2|G| O lal=|G|

Q: 4. Suppose (G₁, *) and (G₂, 0) are groups, let i and J denote the respective identities of G₁ and…

A: and are group. be a group isomorphism.We have to prove that if  has finite order the  has finite…

Q: Let G= How many elements does G have?  To what familiar group is G isomorphic?

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Q: Build the operation table ∗ on G = {e, a, b} knowing that (G, ∗) is a group.

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Q: Let G be a group and let a,b element of G such that (a^3)b = ba. If |a| = 4 and |b| = 2, what is…

A: see below the answer

Q: Give an example of a group G and elements x, y E G with o(xy) o(x)o(y).

A: (a) Give an example of a group G and elements x, y∈G with oxy&lt;minox, oy. Let us take the group…

Q: Let G be a cyclic group ; G=, then (c*b)^=c4* b4 for all a, c, b EG.

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Q: Define ∗ on ℚ+ by a ∗ b =ab/2 . Show that ⟨ℚ+,∗⟩ is a group.

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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: (H,*) is called a of (G,*) if (H,*) is a group.

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Q: Suppose that the piecewise defined function f is defined by 4 - 2x, x≤2 f(x) = { +² +5x-14, x>2…

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Q: Consider the factor group G = R* /Q*. Determine which of the following elements in G are equal. 1Qx,…

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Q: What is the minimum number of generators needed for Z2 ⊕ Z2 ⊕Z2? Find a set of generators and…

A: In this question, we have to find the minimum number of generators needed for the group  ℤ2⊕ℤ2⊕ℤ2…

Q: Let us consider that S is subset of group G such that                      S = { x in G ; x^2=e }…

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Q: 5. Let G be a group. (a) Prove that Inn(G) is a subgroup of Aut (G) (b) Prove that for any € Aut(G)…

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Q: (d) Define * on Q by a * b = ab. Determine whether the binary operation * gives a group on a given…

A: Note: Hi! Thank you for the question as per the honor code, we’ll answer the first question since…

Q: Let G be a group. Assume a and b are elements of G and set H₁ = and H₂ =. Assume |H₁| = 21 and |H₂|…

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Q: Let (G, *) and (H,) be groups. If group (G, *) is cyclic and & H, then (H,) is also cyclic. Select…

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Q: 5. Use a direct proof to show that that in a group G, if (a + b)² = a² * b², then a * b = b * a for…

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Q: (B) Let G be a set of all real numbers excep: . Define * on G by a * b = |a|b, where |a| is the…

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Q: Can you prove that a set is a group, without having an operation? for example can you prove this set…

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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 =&lt;a, b | a2 =e, b2 =e, aba =bab&gt;. To what familiar groupis G isomorphic?

A: Here given that  G =&lt;a, b | a2 =e, b2 =e, aba =bab&gt;. To what familiar group is G isomorphic…

Q: Let a and b belong to a group.   If |a| = 10 and |b| = 21, show that n = {e}

A: Consider a group G. Let a and b be elements of the group G such that a=10 and b=21. Consider the…

Q: 10, Let (G, *) be a group and a, b, c E G. (a) Prove if a *b = a * c, then 6 = C (b) Prove if b * a…

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Q: Let G = &lt;a, b | a6 = b3 = e, b-1ab = a3&gt;. How many elements doesG have? To what familiar group…

A: Given set is G = &lt;a, b | a^6 = b^3 = e, b^-1ab = a^3&gt;.

Q: Let G be a group under multiplication, G be a group under addition and o be an isomorphism from G to…

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Q: Exercise 7.6. Show that each row and column of the group table contains all of the elements of G…

A: Group table and a finite group G.

Q: Suppose the class equation of a group G is |G| = 1 + 4 + 5 + 5 + 5. Show that G has an element of…

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Q: et f be an isomorphism from a grou onto a Group H. et a be an element in G and the orde Fa is 5. it…

A: We have to choose the correct answer from the given option according to the given condition

Q: The alternating group A5 has 5 conjugacy classes, of sizes 1, 12, 12, 15, 20. Use this information…

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Q: Show that every group of order < 60 is solvable.

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Q: A group extension A , G , B of a group A by a group B consists of a group G, an injective group…

A: Given Groups A and B and functions :f:B×B→A, ϕ:B→A at A Normalization Conditions:  ϕ(1)=1…

Q: (1).* 4 and |6| = 2, Let G be a group. Let a, b EG such that |a| and a b = ba. Then determine |ba|,…

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Q: Let a be an element of a group. Suppose abs(a5) = 12. Show that abs(a) = 12 or abs(a) = 60.

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Q: (6) Let (G, *) be a group and a € G. Prove that the order of a is equal to the order of a-1. Prove…

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Q: 4) Let set G be the Cartisian product of G1 × G2, where G1 and G2 are 2 groups. Define a binary…

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Q: 8. Recall that we can define an inner product on the set PC[a, b] as follows: (f.g) = ["f f(x)g(x)…

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Q: Q4: Let A and B be subgroups of a group to that A1 = 4 and 1B1-5- Show that AnB = {e}

A: Let A and B be the two subgroups of Group GSuch  thati.e., the Order of subgroup A is 4 and B is 5…

Q: G is a finite group and f be an automonphism of G satisfying the conditian f(x)= x → x=e for xEG.…

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