True or False If H is a subgroup of the group G then H is on

True or False. If H is a subgroup of the group G then H is one of its cosets in G. Every subgroup of a group G is normal. If |H|

Solution

a) True as He=H where e is identity element.

b) False, only the subgroup N satistfying the property gng^-1 in N for every n in N and g in G( group) are normal.

but if group is abelian then every subgroup is normal.

c) True, as K contain more elements than H it has more subgroups.

d) False, For two distinct cosest identity elements belongs to only one cosets because distinct cosets for set of equivalance classes.

e) yes for each positive integer n we have group for that number.

f) false, because it is only true only if G has order of prime power (sylow thm)