Let G be a finite group. Show that there exists a fixed positive integer n such that an = e for all a in G. (Note that n is independent of a.)

Q: 3. Prove that a subset H of a finite group G is a subgroup of G if and only if a. His nonempty, and…

A: We have to prove given property:

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

A: Let G be a group of order pm, where p is a prime number and m is a positive integer. Then,…

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 a group. Using only the definition of a group, prove that for each a E G, its inverse is…

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Q: Show that in a group G of odd order, the equation x2 =a has aunique solution for all a in G.

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Q: Let x be an element of group G. Prove that if abs(x) = n for some positive integer n, then x-1 =…

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Q: Let G be a finite cyclic group of order 20, and a in G. Then one of the following is possible order…

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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: let G be an abelian Group and let H and K be a susb group of G prove that HK ={hk / h an element H…

A: Consider the provided question, Your question is missing the last statement; I think you want to…

Q: Let G be a group with |G| = pq, where p and q are prime. Prove that every proper subgroup of G is…

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

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Q: Show that if G is a finite group of even order, then there is an a EG such that a is not the…

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Q: Show that if H is any group and h is an element of H, with h" = 1, then there is a unique…

A: Given that H is a group and h ∈H  Now,we define a mapping f:Z→H  such that   f(n) = hn for n∈Z For…

Q: let x be an element of group g. Prove that if |x|=n then x^-1=x^n-1

A: Given 'x' be an element of a group G and |x|=n. As G be a group , inverse of each element of G must…

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 of order n, positive integer, with identity e. Prove or disprove: For any g…

A: Let G be a finite group of order n, with identity e i.e, order of (G) = |G| = n .   Now let g be any…

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 G is a finite group of order n and m is relatively prime to n. If g EG and g™ = e, prove…

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Q: Let S, be the symmetric group and let a be an element of S, defined by: 1 2 3 4 5 67 8 9 ) B = (7…

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Q: If G is a finite group with |G|<120 and G has subgroups of orders 1O, 15 and 20 then the order of G…

A: Using Lagrange's theorem it can be written that if G is a finite group and B is a subgroup of G,…

Q: Let U(n) be the group of units in Zn. If n > 2, prove that there is an element k EU (n) such that k2…

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Q: Let G be a group and let a be a non-identity element of G. Then |a| = 2 if and only if a = a-1.

A: Let G be a group with respect to * . Let e be the identity element,and a is non identity element.…

Q: Let K be a Sylow p-subgroup of a finite group G. Prove that if x ∈ N(K) and the order of x 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: If G is a finite group and some element of G has order equal to the size of G, we ca say that G is:…

A: an abelian group, also called a commutative group, is a group in which the result of applying the…

Q: Let l be an integer greater than 1, and let G be a finite group with no element of order l. Can…

A: Find the attachment for the solution.

Q: for every element a of a ((a) Prove that group 6, 2 (G) is a subset of

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Q: Assume that G is a group such that for all x E G, * x = e. Prove that G is an abelian group.

A: Here we have to prove that G is an abelian group.

Q: An element a of a group G has order n E z+ if and only if a" = e.

A: The given statement is False.

Q: If (G, ) is a group with a = a for all a in G then G is %D abelian

A:

Q: Suppose that a is an element of a group G. Prove that if there is some integer n, n notequalto 0,…

A: Suppose that a is an element of a group G. Prove that if there is some integer n, such that n≠0 and…

Q: Prove that every subgroup of Z is either the trivial group, {0}, or nZ = {nx | x E Z} for some n E…

A: To prove: That every subgroup of ℤ is either the trivial group{0} or nℤ=nxx∈ℤfor some n∈ℕ. Proof:…

Q: Prove that if G is a finite group, then the index of Z(G) cannot be prime.

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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: Let G be a group with p or more distinct elements of order p for some prime p. Show that G is not…

A: Given : Let G be a group which has p  elements of order p, p is a prime no.  To show G is not cyclic…

Q: Let G be a group and H ≤G. Let x ∈G be an element of finite order n. Prove that if k is the least…

A: It is given that, H is a subgroup of G and |x|=n. Also  k is the least positive integer such that…

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: Let G be a group. Prove that Z(G) < G.

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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: '. Assume that G is a group such that for all x E G, x * x = e. Prove that G is an abelian group.

A: Consider any two elements a and b in G. So, a,b,ab,ba∈G. Note that I am directly writing the…

Q: Let G be a group and a e G such that o(a) = n < oo. Show that a = a' if and only if k =l mod n. %3D

A: Let G be a group and a∈G such that Oa=n&lt;∞. Show that ak=al if and only if k≡l mod n. If k=l the…

Q: Let G be a finite group, prove that there exists m E G such that a ^ m = e for each a E G and where…

A: Let G be a finite group, prove that there exists m E G such that a ^ m = e for each a E G and where…

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: 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: Let G be a group such that a^2 = e for each aEG. Then G is * Non-abelian Cylic Finite Abelian

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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: If A is an abelian group with A <G and B is any subgroup of G, prove that ANB < AB.

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Q: Let G be a finite non-abelian simple group and let q be prime, then [G] is

A: It is given that G be any finite non Abelian simple group and q be any prime. We have to determine…

Q: Let G be a finite group. Show that there exist a positive integer m such that for each gE G, gm = e.

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Q: Let G, H, and K be finitely generated abelian groups. Prove or disprove: If G × H ∼= K × G, then H…

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