Is there a proper subgroup of the group Q that has finite index?

Deduce that QZ has no proper subgroups of finite index. n = q is an element of N. Hence N = Q.

Are all subgroups finite?

All subgroups of Zp∞ are finite If the set of the orders of elements of H is infinite, then for all element z∈Zp∞ of order pk, there would exist an element z′∈H of order pk′>pk. Therefore the set of the orders of the elements of H is finite.

Is Q za finite group?

There are many infinite groups with this property that every element of the group has a finite order; for example, any direct product of infinitely many copies of a finite group.

What is the index of a Coset?

The index of H in G, denoted [G : H], is equal to the number of left cosets of H in G. Note that even though G might be infinite, the index might still be finite. For example, suppose that G is the group of integers and let H be the subgroup of even integers.

Does the intersection of two finite index subgroups have finite index?

Another way to see that the answer is “yes”: In this thread it is shown that any finite index subgroup of G contains a subgroup which is normal and of finite index in G.

How do you find proper subgroups?

The trivial subgroup of any group is the subgroup {e} consisting of just the identity element. A proper subgroup of a group G is a subgroup H which is a proper subset of G (that is, H ≠ G). This is usually represented notationally by H < G, read as “H is a proper subgroup of G”.

Can subgroups be infinite?

An infinite cyclic group has infinitely many subgroups. Therefore, all cyclic subgroups of G are finite.

How many subgroups does Z20 have?

(e) Draw the subgroup lattice of Z20 [Note: 20 = 22 · 5]. We know that there is exactly one subgroup per divisor of 20. These subgroups are arranged ac- cording to divisibility, so to draw a subgroup lat- tice we should first draw a divisibility lattice for the divisors of 20.

Is every subgroup of an Abelian group Abelian?

Every subgroup of an abelian group is normal, so each subgroup gives rise to a quotient group. Subgroups, quotients, and direct sums of abelian groups are again abelian. The finite simple abelian groups are exactly the cyclic groups of prime order.