Pool problems in group theory

Let G be a group. Prove that G is non-cyclic if and only if G is the union of its proper subgroups.


Let G be a group, and G×G the direct product. The set D={(g,g,)gG} is a subgroup of G×G. Prove that if D is normal in G×G then G is abelian.


The dihedral group, D8, is the group of eight rigid symmetries of a square. Prove that D8 is not the internal direct product of two of its proper subgroups.


Let G be a finite group and H,KG be normal subgroups of relatively prime order. Prove that G is isomorphic to a subgroup of G/H×G/K.


Suppose G is a group that contains normal subgroups H,KG with HK={e} and HK=G. Prove that GH×K.


Let G be the group of upper-triangular real matrices [ab0d] with a,d0, under matrix multiplication. Let S be the subset of G defined by d=1. Show that S is normal and that G/SR×, the multiplicative group of nonzero real numbers.


Let G be a group and suppose Aut(G) is trivial.

  1. Show that G is abelian.
  2. Show that for any abelian group H, the inversion map ϕ(h)=h1 is an automorphism.
  3. Use parts (1) and (2) above to show that g2 is the identity element for every gG.

  1. Suppose N is a normal subgroup of a group G and πN:GG/N is the usual projection homomorphism, defined by πN(g)=gN. Prove that if ϕ:GH is any homomorphism with Nker(ϕ), then there exists a unique homomorphism ψ:G/NH such that ϕ=ψπN. (You must explicitly define ψ, show it is well defined, show ϕ=ψπN, and show that ψ is uniquely determined.)
  2. Prove the:
    Third Isomorphism Theorem. If M,NG with NM, then (G/N)/(M/N)G/M.

Let G be a group and suppose Aut(G) is trivial.

  1. Show that G is abelian.
  2. Show that for any abelian group H, the inversion map ϕ(h)=h1 is an automorphism.
  3. Use parts (1) and (2) above to show that g2 is the identity element for every gG.

Let G be a group and aG be an element. Let nN be the smallest positive number such that an=e, where e is the identity element. Show that the set

{e,a,a2,,an1}

contains no repetitions.


Let G be a finite abelian group of odd order. Let ϕ:GG be the function defined by ϕ(g)=g2 for all gG. Prove that ϕ is an automorphism.


Let G be a group with exactly two conjugacy classes. Prove that G is abelian, and describe all such groups (with proof).


Let Zn denote the cyclic group of order n. Suppose mN is relatively prime to n. Define the function μm:ZnZn by m[a]n=[ma]n.

  1. Prove that the map μm is a well-defined automorphism of Zn.
  2. Prove that any automorphism of Zn has the form μm for some m.

Let G be a finite group and n>1 an integer such that (ab)n=anbn for all a,bG. Let

Gn={cGcn=e}andGn={cncG}

You may take for granted that these are subgroups. Prove that both Gn and Gn are normal in G, and |Gn|=[G:Gn].


Suppose G is a group, HG a subgroup, and a,bG. Prove that the following are equivalent:

  1. aH=bH
  2. baH
  3. b1aH

Let G be a group, and let Aut(G) denote the group of automorphisms of G. There is a homomorphism γ:GAut(G) that takes sG to the automorphism γs defined by γs(t)=sts1.

  1. Prove rigorously, possibly with induction, that is γs(t)=tb, then γsn(t)=tbn.
  2. Suppose sG has order 5, and sts1=t2. Find the order of t. Justify your answer.

Let G be an abelian group and GT be the set of elements of finite order in G.

  1. Prove that GT is a subgroup of G.
  2. Prove that every non-identity element of G/GT has infinite order.
  3. Characterize the elements of GT when G=R/Z, where R is the additive group of real numbers.

Suppose G is a finite group of even order.

  1. Prove that an element in G has order dividing 2 if and only if it is its own inverse.
  2. Prove that the number of elements in G of order 2 is odd.
  3. Use (2) to show G must contain a subgroup of order 2.

Let N be a finite normal subgroup of G. Prove there is a normal subgroup M of G such that [G:M] is finite and nm=mn for all nN and mM.

Hint: You may use the fact that the centralizer C(h):={gGghg1=h} is a subgroup of G.)


Suppose G is a nonempty finite set that has an associative pairing G×GG, written (x,y)xy. Suppose this pairing satisfies left and right cancellation: xy=xy implies y=y, and xy=xy implies x=x. Prove there exists an element e of G such that for all xG, ex=xe=x. Justify your reasoning as carefully as possible.


Show that every finite group with more than two elements has a nontrivial automorphism.


Let σ:GH be a group epimorphism. Let N be a normal subgroup of G and K=σ(N), the image of N in H.

  1. Prove that K is a normal subgroup of H. Give an example to show that this is not true if σ is not onto.
  2. Under what conditions does σ induce a homomorphism G/NH/K, and when is this an isomorphism? Prove your answer.

Suppose G1 and G2 are groups, with identity elements e1 and e2, respectively. Prove that if ϕ:G1G2 is an isomorphism, then ϕ(e1)=e2.


Suppose A and B are subgroups of a group G, and suppose B is of finite index in G.

  1. Show that the index of ABA is finite, and in fact |A:AB||G:B|. Hint: Find a set map A/ABG/B.
  2. Prove that equality holds in (a) if and only if G=AB.

Let G be a group. For each aG, let γa denote the automorphism of G defined by γa(b)=aba1 for all bG. The set Inn(G)={γa:aG} is a subgroup of the automorphism group of G, called the subgroup of inner automorphisms.

Prove that Inn(G) is isomorphic to G/Z(G), where Z(G) is the center of G.


Let G be a group of order 2p, where p is an odd prime. Prove G contains a nontrivial, proper normal subgroup.


Prove from the definition along that there are no nonabelian groups of order less than 5.


Let G be a group and H,KG be normal subgroups with HK={e}. Show that each element in H commutes with every element in K.


Let G be a group and N a normal subgroup of G. Let aN denote the left coset defined by aG, and consider the binary operation

G/N×G/NG/N

given by (aN,bN)abN.

  1. Show the operation is well defined.
  2. Show the operation is well defined only if the subgroup N is normal.

Let G be a group, HG a subgroup that is not normal. Prove there exist cosets Ha and Hb such that HaHbHab.


Let H be a subgroup of a group G. The normalizer of H in G is the set NG(H)={gGgH=Hg}.

  1. Prove NG(H) is a subgroup of G containing H.
  2. Prove NG(H) is the largest subgroup of G in which H is normal.

Let G be a group and suppose HG. The normalizer of H in G is defined to be N(H)={gG|gH=Hg} and the centralizer of H in G is defined to be C(H)={gG|gh=hg for all hH}.

  1. Prove that N(H) is a subgroup of G.
  2. Prove that C(H) is a normal subgroup of N(H) and that N(H)/C(H) is isomorphic to a subgroup of Aut(H).

Suppose G is a cyclic group of order n, and tG is a generator.

  1. Give a positive integer d such that t1=td.
  2. Let c be an integer and let m=gcd(n,c). Prove that the order of tc is nm.

Let G be a finite group. Prove from the definitions that there exists a number N such that aN=e for all aG.


Suppose G is a group and NG is a finite normal subgroup. Prove that if G/N contains an element of order n, then G also contains an element of order n.


Suppose ϕ:GG is a surjective homomorphism, HG is a subgroup containing ker(ϕ), and H=ϕ(H). Prove ϕ1(H)=H, where ϕ1(H)={gGϕ(g)H}. Make sure to state explicitly where each hypothesis is used.


Let G be a group, and H,K be subgroups of G. Let HK={hkhH,kK} denote the set product. Prove that HK is a group if and only if HK=KH.


Suppose G is a nontrivial finite group and H,KG are normal subgroups with gcd(|H|,|K|)=1.

  1. Define a nontrivial group homomorphism ϕ:GG/H×G/K
  2. Prove G is isomorphic to a subgroup of G/H×G/K.
  3. Suppose gcd(m,n)=1. Prove ZmnZm×Zn.

Suppose G is a group, H and K are normal subgroups of G, and HK.

  1. Define a group homomorphism from K to G/H.
  2. Compute the kernel of the homomorphism in (a), and apply the First Isomorphism Theorem.

Let G be a finite group and Z(G) denote its center.

  1. Prove that if G/Z(G) is cyclic, then G is abelian.
  2. Prove that if G is nonabelian, then |Z(G)|14|G|.

Let G be a group, mN, and gG be an element such that gm=e. Prove that o(g)m, where o(g) is the order of g.


  1. Show that if G is any group (not necessarily finite) and H is a subgroup, then G is a disjoint union of left cosets of H.
  2. State and prove Lagrange's Theorem for finite groups.

Let G be a group and HG a subgroup. For each coset aH of H in G, define the set

GaH={bG|baH=aH}.

  1. Prove that GaH is a subgroup of G.
  2. Suppose that H is normal in G. Prove that GaH=H.

Let G be a group of order 2n for some positive integer n>1.

  1. Prove there exists a subgroup K of G of order 2.
  2. Suppose K in (a) is a normal subgroup. Prove that K is contained in the center Z(G). (Recall Z(G)={aGab=ba for all bG}.)

The additive group Z=(Z,+) of rational integers is a subgroup of the additive group Q=(Q,+). Show that Q has infinite index in Q.


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