0

# G is finite where the number of subgroups in g is finite?

Updated: 10/23/2022

Wiki User

14y ago

Actually a stronger statement can be made:
A group G is finite if and only if the number of its subgroups is finite

Let G be a group. If G is finite there is only a finite number of subsets of G, so clearly
a finite number of subgroups.
Now suppose G is infinite , let's
suppose one element has infinite order. The this element generates an infinite cyclic
group which in turn contains infinitely many subgroups.

Now suppose all the subgroups have finite order Take some element of G and let it generate a finite group H. Now take another element of G not in H and let it generate a finite group I. Keep doing this by next picking an element of G not H or I. You can continue this way.

Wiki User

14y ago

Earn +20 pts
Q: G is finite where the number of subgroups in g is finite?
Submit
Still have questions?
Related questions

### G is finite group if and only if the number of it is subgroups is finite?

False. G may be a finite group without sub-groups.

### How do you prove that order of a group G is finite only if G is finite and vice versa?

(1). G is is finite implies o(G) is finite.Let G be a finite group of order n and let e be the identity element in G. Then the elements of G may be written as e, g1, g2, ... gn-1. We prove that the order of each element is finite, thereby proving that G is finite implies that each element in G has finite order. Let gkbe an element in G which does not have a finite order. Since (gk)r is in G for each value of r = 0, 1, 2, ... then we conclude that we may find p, q positive integers such that (gk)p = (gk)q . Without loss of generality we may assume that p> q. Hence(gk)p-q = e. Thus p - q is the order of gk in G and is finite.(2). o(G) is finite implies G is finite.This follows from the definition of order of a group, that is, the order of a group is the number of members which the underlying set contains. In defining the order we are hence assuming that G is finite. Otherwise we cannot speak about quantity.Hope that this helps.

### How do we prove that a finite group G of order p prime is cyclic using Lagrange?

Lagrange theorem states that the order of any subgroup of a group G must divide order of the group G. If order p of the group G is prime the only divisors are 1 and p, therefore the only subgroups of G are {e} and G itself. Take any a not equal e. Then the set of all integer powers of a is by definition a cyclic subgroup of G, but the only subgroup of G with more then 1 element is G itself, therefore G is cyclic. QED.

### What is the cardinal number of Set G if G 1 3 5 7?

The cardinality of a set is its size. For instance, since the set G contains 4 elements, then its cardinality is 4. So if the set has a finite number of elements (meaning it is a finite set), you can find its cardinality, otherwise you cannot (meaning it is an infinite set).

### Show that if G is a finite group with identity 'e' and with an even number of elements then there is 'a' not equal 'e' in G that xx equal 'e'?

This follows immediately from the first Sylow theorem.

### What has the author Richard G Swan written?

Richard G. Swan has written: 'K-theory of finite groups and orders' -- subject(s): Finite groups, K-theory

### What has the author G Nebe written?

G. Nebe has written: 'Finite rational matrix groups' -- subject(s): Finite groups, Forms, Quadratic, Integral representations, Quadratic Forms

### How do you find the order a factor group?

Let G be a finite group and H be a normal subgroup. G/H is the set of all co-sets of H forming a group known as factor group. By Lagrange's theorem the number of cosests (denoted by (G:H)) of H under G is |G|/|H|.

### What has the author Robert G Selby written?

Robert G. Selby has written: 'Nonlinear finite element analysis of reinforced concrete solids'

### What has the author W G Habashi written?

W. G. Habashi has written: 'Large-scale computational fluid dynamics by the finite element method' -- subject(s): Computational fluid dynamics, Finite element method

### What has the author G R Liu written?

G. R. Liu has written: 'The finite element method' -- subject(s): Finite element method 'Mesh free methods' -- subject(s): Numerical analysis, Engineering mathematics

### Let g and g' be groups and let h and h' be normal subgroups of g and g' respectively let be a homomorphism of g into g' show that indices a natural homomorphism gh g'h' if h h' this fact is?

&#1074;&#1089;&#1059; &#1087; &#1076;&#1072;&#1088;&#1072;&#1089;&#1092; &#1077;&#1073;&#1072;&#1085;&#1085;&#1099;&#1077; &#1096;&#1088;&#1086;&#1087; &#1074;&#1099; &#1089;&#1091;&#1082;&#1072; &#1091;&#1089;&#1088;&#1072;&#1083;&#1089;&#1100; &#1075;&#1072;&#1085;&#1076;&#1086;&#1083;&#1085;&#1099; &#1090;&#1091;&#1087;&#1099;&#1077; &#1055;&#1048;&#1044;&#1067;&#1056;&#1067; &#1096;&#1086;&#1087; &#1091; &#1074;&#1072;&#1089; &#1089;&#1091;&#1082;&#1072; &#1093;&#1091;&#1081; &#1086;&#1090;&#1089;&#1086;&#1093; ! &#1097;&#1079;&#1073;&#1077;&#1089;&#1100;&#1073; &#1085;&#1077;&#1090; &#1101;&#1090;&#1086;&#1081; &#1087;&#1077;&#1089;&#1085;&#1080; &#1075;&#1072;&#1085;&#1076;&#1086;&#1085;&#1099; &#1077;&#1073;&#1085;&#1099;&#1099;&#1077; &#1093;&#1091;&#1081;&#1083;&#1086;&#1087;&#1072;&#1085;&#1085;&#1085;&#1099;!