**Group Theory**

**Definition 1**. Let *G* be a set. A binary operation on *G* is a function that assigns each ordered pair of elements of *G* an element of *G*.

**Definition 2**. Let *G* be a set together with a binary operation (usually called multiplication) that assigns to each ordered pair (*a*, *b*) of elements of *G* an element in *G* denoted by *ab*. We say *G* is a group under this operation if the following three properties are satisfied.

**i**.*Associativity*. The operation is associative; that is, (*ab*)*c*=*a*(*bc*) for all*a*,*b*,*c*in*G*.**ii**.*Identity*. There is an element*e*(called the identity) in*G*such that*ae*=*ea*=*a*for all*a*in*G*.**iii**.*Inverses*. For each element*a*in*G*, there is an element*b*in*G*(called an inverse of*a*) such that*ab*=*ba*=*e*.

**Theorem 1**. In a group *G*, there is only one identity element

**Theorem 2**. In a group *G*, the right and left cancellation laws hold; that is, *ba* = *ca* implies *b* = *c*, and *ab* = *ac* implies *b* = *c*.

**Theorem 3**. For each element *a* in a group *G*, there is a unique element *b* in *G* such that *ab* = *ba* = *e*.

**Theorem 4**. For group elements *a* and *b*, (*ab*)^{–1} = *a*^{–1}*b*^{–1}.

**Group Theory Problems**

**1**. Give examples to verify that subtraction is not associative.

**2**. Determine if the following sets *G* with the operation indicated form a group. If not, point out which of the group axioms fail.

- (a)
*G*= set of all integers,*a***b*=*a*–*b*. - (b)
*G*= set of all integers,*a***b*=*a*+*b*+*ab.* - (c)
*G*= set of nonnegative integers,*a***b*=*a*+*b*. - (d)
*G*= set of all rational numbers ≠ –1,*a***b*=*a*+*b*+*ab*. - (e)
*G*= set of all rational numbers with denominator divisible by 5 (written so that numerator and denominator are relatively prime),*a***b*=*a*+*b*. - (f)
*G*a set having more than one element,*a***b*=*a*for all*a*,*b*∈*G*.

**3**. Give two reasons why the set of odd integers under addition is not a group.

**4**. Prove that the set of all 2 × 2 matrices with entries from ℝ and determinant +1 is a group under matrix multiplication.

**5**. Show that {1, 2, 3} under *multiplication* *modulo* 4 is not a group but that {1, 2, 3, 4} under *multiplication modulo* 5 is a group.

**6**. An abstract algebra teacher intended to give a typist a list of nine integers that form a group under *multiplication* *modulo* 91. Instead, one of the nine integers was inadvertently left out, so that the list appeared as 1, 9, 16, 22, 53, 74, 79, 81. Which integer was left out?

**7**. Give an example of a group with 105 elements. Give two examples of groups with 44 elements.

**8**. Give an example of group elements *a* and *b* with the property that *a*^{–1}*ba* ≠ *b*.

**9**. List the members of *H* = {*x*^{2} | *x* ∈ *D*_{4}} and *K* = {*x* ∈ *D*_{4} | *x*^{2} = *e*}.

**10**. (*Law of exponents for Abelian groups*) Let *a* and *b* be elements of an Abelian group and let *n* be any integer. Show that (*ab*)* ^{n}* =

*a*. Is this also true for non-Abelian groups?

^{n}b^{n}**11**. Prove that a group *G* is Abelian if and only if (*ab*)^{–1} = *a*^{–1}*b*^{–1} for all *a* and *b* in *G*.

**12**. The integers 5 and 15 are among a collection of 12 integers that form a group under multiplication modulo 56. List all 12.

**13**. Prove that every group table is a Latin square (that is, each element of the group appears exactly once in each row and each column.

**14**. Suppose the table below is a group table. Fill in the blank entries.

**15**. Prove that if (*ab*)^{2} = *a*^{2}*b*^{2} in a group *G*, then *ab* = *ba*.

**16**. Prove that the set of all rational numbers of the form 3* ^{m}*6

*, where*

^{n}*m*and

*n*are integers, is a group under multiplication.

**17**. Let *a*, *b*, and *c* be elements of a group. Solve the equation *axb* = *c* for *x*. Solve *a*^{–1}*xb* = *c* for *x*.

**18**. Let *G* be a group with the following property: Whenever *a*, *b*, and *c* belong to *G* and *ab* = *ca*, then *b* = *c*. Prove that *G* is Abelian (*cross cancellation* implies commutativity).

**19**. Give an example of a group with elements *a*, *b*, *c*, *d*, and *x* such that *axb* = *cxd* but *ab* ≠ *cd* (hence *middle cancellation* is not valid in groups).

**20**. Suppose that *G* is a group with the property that for every choice of elements in *G*, *axb* = *cxd* implies *ab* = *cd*. Prove that *G* is Abelian (*middle cancellation* implies commutativity).

**21**. Let *G* be a set with an operation * such that:

**i**. *G* is closed under *.

**ii**. * is associative.

**iii**. There exists an element *e* ∈ *G* such that *e* * *x* = *x* for all *x* ∈ *G*.

**iv**. Given *x* ∈ *G*, there exists a *y* ∈ *G* such that *y* * *x* = *e*.

Prove that *G* is a group (thus you must show that *x* * *e* = *x* and *x* * *y* = *e* for *e*, *y* as above).

**22**. Let *G* be a finite nonempty set with an operation * such that:

**i**. *G* is closed under *.

**ii**. * is associative.

**iii**. Given *a*, *b*, *c* ∈ *G* with *a* * *b* = *a* * *c*, then *b* = *c*.

**iv**. Given *a*, *b*, *c* ∈ *G* with *b* * *a* = *c* * *a*, then *b* = *c*.

Prove that *G* must be a group under *. Give an example to show that the result can be false if *G*

is an infinite set.

**23**. If *G* is a group in which (*ab*)* ^{i}* =

*a*for three consecutive integers

^{i}b^{i}*i*, prove that

*G*is abelian. Show that the result of would not always be true if the word

*three*were replaced by

*two*.

**24**. Let *G* be a group in which (*ab*)^{3} = *a*^{3}*b*^{3} and (*ab*)^{5} = *a*^{5}*b*^{5} for all *a*, *b* ∈ *G*. Show that *G* is abelian.

**25**. Let *R* be any rotation in some dihedral group and *F* any reflection in the same group. Prove that *RFR* = *F*.

**26**. If *A*, *B* are subgroups of *G*, show that *A* ∩ *B* is a subgroup of *G*.

**27**. What is the cyclic subgroup of ℤ generated by –1 under +?

**28**. Let *S*_{3} be the symmetric group of degree 3. Find all the subgroups of *S*_{3}.

**29**. Verify that *Z*(*G*), the centre of *G*, is a subgroup of *G*.

**30**. Prove that a cyclic group is abelian.

**31**. If *G* is cyclic, show that every subgroup of *G* is cyclic.

**32**. If *G* has no proper subgroups, prove that *G* is cyclic.

**33**. If *G* has no proper subgroups, prove that *G* is cyclic of order *p*, where *p* is a prime number.

**34**. If *A*, *B* are subgroups of an abelian group *G*, let *AB* = {*ab* | *a* ∈ *A*, *b* ∈ *B*}. Prove that *AB* is a subgroup of *G*.

**35**. Give an example of a group *G* and two subgroups *A*, *B* of *G* such that *AB* is not a subgroup of *G*.

**36**. If *A*, *B* are subgroups of *G* such that *b*^{–1}*Ab* ⊂ *A* for all *b* ∈ *B*, show that *AB* is a subgroup of *G*.

**37**. If *A* and *B* are finite subgroups, of orders *m* and *n*, respectively, of the abelian group *G*, prove that *AB* is a subgroup of order *mn* if *m* and *n* are relatively prime.

**38**. Show that any group of order 4 or less is abelian.

**39**. Show that a group of order 5 must be abelian.

**40**. If *G* is a group in which *a*^{2} = *e* for all *a* ∈ *G*, show that *G* is abelian.

**41**. If *G* is a finite group of even order, show that there must be an element *a* ≠ *e* such that *a* = *a*^{–1}.

**42**. If *G* is a finite group, prove that, given *a* ∈ *G*, there is a positive integer *n*, depending on *a*, such that *a ^{n}* =

*e*. Also show that there is an integer

*m*> 0 such that

*a*=

^{m}*e*for all

*a*∈

*G*.

**43**. Let *G* be a finite group. Show that the number of elements *x* of *G* such that *x*^{3} = *e* is odd. Show that the number of elements *x* of *G* such that *x*^{2} ≠ *e* is even.

**44**. Let *S* be a set having an operation * which assigns an element *a* * *b* of *S* for any *a*, *b* ∈ *S*. Let us assume that the following two rules hold:

**i**. If *a*, *b* are any objects in *S*, then *a* * *b* = *a*.

**ii**. If *a*, *b* are any objects in *S*, then *a* * *b* = *b* * *a*.

Show that *S* can have at most one object.

**45**. Show that a set having *n* elements has 2* ^{n}* subsets. (b) If 0 <

*m*<

*n*, how many subsets are there that have exactly

*m*elements?